diff --git "a/89FJT4oBgHgl3EQfoCxt/content/tmp_files/load_file.txt" "b/89FJT4oBgHgl3EQfoCxt/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/89FJT4oBgHgl3EQfoCxt/content/tmp_files/load_file.txt" @@ -0,0 +1,1991 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf,len=1990 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11594v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='FA] 27 Jan 2023 ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS GERHARD SCHINDL Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' N-functions and their growth and regularity properties are crucial in order to in- troduce and study Orlicz classes and Orlicz spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We consider N-functions which are given in terms of so-called associated weight functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' These functions are frequently appearing in the theory of ultradifferentiable function classes and in this setting additional information is available since associated weight functions are defined in terms of a given weight sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We express and characterize several known properties for N-functions purely in terms of weight sequences which allows to construct (counter)-examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Moreover, we study how for abstractly given N- functions this framework becomes meaningful and finally we establish a connection between the complementary N-function and the recently introduced notion of the so-called dual sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Introduction Let us start by recalling briefly the basic definitions of Orlicz classes and Orlicz spaces, we refer to [11] and to the informative summary presented in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' For this let F be a so-called N-function, see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1 below and [11, Chapter 1, §1, 3], [1, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Moreover, let Ω be a bounded and closed set in Rd and consider on Ω the usual Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then the Orlicz class is given by LF (Ω) := {u : Ω → R, measurable : � Ω F(u(x))dx < +∞}, and the Orlicz space by L∗ F (Ω) := {u : Ω → R, measurable : � Ω u(x)v(x)dx < +∞, ∀ v ∈ LF c(Ω)}, with F c denoting the so-called complementary N-function which is again an N-function, see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1 and [11, Chapter 1, §2], [1, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' If we do not need to specify the set Ω we omit it and only write LF resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' L∗ F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We mention that sometimes in the literature slightly different assumptions on F are used, see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In order to study these classes several growth and regularity assumptions for F and F c are considered frequently in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Most prominent are the so-called ∆2, ∆3, ∆2 and ∆′ condition for F, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' [11, Chapter I, §4-§6], [1, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2] and Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' If F c satisfies a ”∆-type” property then by convention usually one writes that F has the corresponding ”∇-type” condition (and vice versa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The aim of this paper is to introduce and study N-functions FM which are given in terms of a given sequence M ∈ RN >0, see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='13, via the so-called associated weight function ωM (see Date: January 30, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 26A12, 26A48, 26A51, 46E30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Orlicz classes and Orlicz spaces, N-functions, associated weight functions, weight se- quences, dual sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Schindl is supported by FWF-Project P33417-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 1 2 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' SCHINDL Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Here the sequence is expressing the growth of FM and M is assumed to satisfy mild standard and growth assumptions, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Recall that ωM is appearing frequently in the theory of classes of ultradifferentiable (and ultraholomorphic) functions defined in terms of weight sequences and it serves also as an example for an abstractly given weight function ω in the sense of Braun-Meise-Taylor, see [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Consequently, FM contains additional information expressed in the underlying sequence M and the idea is to exploit this fact, to ”combine” the ultradifferentiable-type and the Orlicz-type setting and to treat the following questions/problems: (∗) Study for FM the aforementioned known and important growth properties for abstractly given N-functions in terms of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' When given two N-functions FM, FL expressed in terms of sequences M and L, then study the crucial relation between N-functions (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4)) in terms of a growth comparison between M and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) Use this knowledge in order to construct (counter)-examples illustrating the relations and connections between the different growth conditions for N-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) Compare the (partially) new growth properties for weight sequences with known conditions appearing in the ultradifferentiable setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) Check if these properties and conditions can be transferred from given M, L to related constructed sequences, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' the point-wise product M · L and the convolution product M ⋆ L (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) Let G be an abstractly given N-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Is it then possible to associate with G a weight sequence, say M G, and to apply the derived results in order to get information for G itself (via using FMG)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) When given M and FM study and establish the connection between the notions of the com- plementary N-function F c M and the dual sequence D (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' M) which has been introduced in [5, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='40, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' This question has served as the main motivation for writing this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (The relevance of D is given by the fact that the so-called orders and Matuszewska indices for M and D are ”reflected/inverted” as it has been shown in the main result [5, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Concerning these notions we refer to [5, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2], [7] and the citations there for more details and precise definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=') However, it turns out that in the weight sequence setting we cannot expect that the relevant function t �→ ϕωM (t) := ωM(et) (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='12)) directly is an N-function, see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6 for more explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' One can overcome this technical problem by using the fact that ϕωM is the so-called principal part of an N-function FM, see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' On the other hand we mention that ϕωM also allows to compare different used notions for being a weight in the ”Orlicz-setting”, see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='14 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Note that the crucial conditions for M in order to ensure the desired growth properties for FM are partially (slightly) different compared with the known ones used in the ultradifferentiable setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' This is mainly due to the fact that the relevant function under consideration is given by ϕωM and not by ωM directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' For example, the prominent ∆2-property for N-functions (see Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1) is also appearing as a known growth condition in the ultradifferentiable weight function setting (abbreviated by (ω1) in this work) but the crucial condition for M is different (see Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2 and the comments there).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The paper is structured as follows: In Section 2 all relevant definitions concerning weight sequences and (associated) weight functions are given and in Section 3 we recall and introduce the notions of (associated) N-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In Section 4 we focus on the study of the comparison between associated ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS 3 N-functions, see Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4, and give in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2 several sufficient conditions on the sequences to ensure equivalence between the associated N-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Section 5 is dedicated to the study of the meaning of the associated weight sequence M G when G is an abstractly given N-function, see Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In Section 6 we introduce and study the complementary N-function F c M (see Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4 and Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5) and establish the connection between F c M and the dual sequence D, see the main statement Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Finally, in Section 7 we provide a detailed study of growth and regularity conditions for N-functions in the weight sequence setting, see Theorems 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='12, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='18 and Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Some (counter)-examples and their consequences are mentioned as well, see (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='16) and Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Weights and conditions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Weight sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We write N = {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='} and N>0 := {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Given a sequence M = (Mj)j ∈ RN >0 we also use µ = (µj)j defined by µj := Mj Mj−1 , µ0 := 1, and analogously for all other appearing sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' M is called normalized if 1 = M0 ≤ M1 holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' For any ℓ > 0 we put M ℓ := (M ℓ j )j∈N, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' the ℓ-th power, and write M · L = (MjLj)j∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Finally, let us introduce the convolved sequence M ⋆ L by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1) M ⋆ Lj := min 0≤k≤j MkLj−k, j ∈ N, see [10, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='15)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' M is called log-convex if ∀ j ∈ N>0 : M 2 j ≤ Mj−1Mj+1, equivalently if (µj)j is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' If M is log-convex and normalized, then both j �→ Mj and j �→ (Mj)1/j are non-decreasing and (Mj)1/j ≤ µj for all j ∈ N>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' M (with M0 = 1) has condition moderate growth, denoted by (mg), if ∃ C ≥ 1 ∀ j, k ∈ N : Mj+k ≤ Cj+kMjMk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In [10] this is denoted by (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2) and also known under the name stability under ultradifferential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' For our purpose it is convenient to consider the following set of sequences LC := {M ∈ RN >0 : M is normalized, log-convex, lim j→+∞(Mj)1/j = +∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We see that M ∈ LC if and only if 1 = µ0 ≤ µ1 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' , limj→+∞ µj = +∞ (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' [15, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 104]) and there is a one-to-one correspondence between M and µ = (µj)j by taking Mj := �j k=0 µk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' If M, L ∈ LC, then M · L, M ⋆ L ∈ LC (for the convolution see [10, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M, L ∈ RN >0 be given, then write M ≤ L if Mj ≤ Lj for all j ∈ N and M ⪯ L if supj∈N>0 � Mj Lj �1/j < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Sequences M and L are called equivalent, denoted by M ≈ L, if M⪯L and L⪯M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Frequently we will consider the following important examples belonging to the class LC: (i) The Gevrey-sequences Gs, s > 0, given by Gs j := j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) The sequences M q,n, q, n > 1, given by M q,n j := qjn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' If n = 2, then M q,2 is the so-called q-Gevrey-sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 4 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' SCHINDL 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Associated weight function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M ∈ RN >0 (with M0 = 1), then the associated function ωM : R → R ∪ {+∞} is defined by ωM(t) := sup j∈N log �|t|j Mj � for t ∈ R, t ̸= 0, ωM(0) := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' For an abstract introduction of the associated function we refer to [12, Chapitre I], see also [10, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Note that ωM is here extended to whole R in a symmetric (even) way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' If lim infj→+∞(Mj)1/j > 0, then ωM(t) = 0 for sufficiently small t, since log � tj Mj � < 0 ⇔ t < (Mj)1/j holds for all j ∈ N>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (In particular, if Mj ≥ 1 for all j ∈ N, then ωM is vanishing on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=') Moreover, under this assumption t �→ ωM(t) is a continuous non-decreasing function, which is convex in the variable log(t) and tends faster to infinity than any log(tj), j ≥ 1, as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' limj→+∞(Mj)1/j = +∞ implies that ωM(t) < +∞ for each finite t which shall be considered as a basic assumption for defining ωM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' For given M ∈ LC we define the counting function ΣM : [0, +∞) → N by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2) ΣM(t) := |{j ∈ N>0 : µj ≤ t}|, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' ΣM(t) is the maximal positive integer j satisfying µj ≤ t (and ΣM(t) = 0 for 0 ≤ t < µ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' It is known that ωM and ΣM are related by the following integral representation formula, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' [12, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' III] and [10, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11)]: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3) ωM(t) = � t 0 ΣM(u) u du = � t µ1 ΣM(u) u du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Consequently, ωM vanishes on [0, µ1], in particular on the unit interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' By definition of ωM the following formula is immediate: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4) ∀ ℓ > 0 ∀ t ≥ 0 : ℓωM(t1/ℓ) = ωMℓ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In [10, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5] for given M, L ∈ LC it is shown that ∀ t ≥ 0 : ΣM⋆L(t) = ΣM(t) + ΣL(t), which implies by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3) ∀ t ≥ 0 : ωM⋆L(t) = ωM(t) + ωL(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Finally, if M ∈ LC, then we can compute M by involving ωM as follows, see [12, Chapitre I, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8] and also [10, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2]: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5) Mj = sup t≥0 tj exp(ωM(t)), j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M ∈ LC be given, we comment on the surjectivity of ΣM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) Obviously ΣM(t) ∈ N for all t ≥ 0 and ΣM is surjective if and only if µj < µj+1 for all j ∈ N>0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' if j �→ µj is strictly increasing: In this case we have ΣM(t) = j for all µj ≤ t < µj+1, j ∈ N>0, and ΣM(t) = 0 for t ∈ [0, µ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) Note that µj < µj+1 for all j does not hold automatically for all sequences belonging to the set LC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' However, when given M ∈ LC, then we can always find � M ∈ LC such that M and � M are equivalent and such that the corresponding sequence of quotients (�µj)j≥1 is strictly increasing, see [6, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' This formal switch allows to avoid technical complications resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' to simplify arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS 5 More precisely, in [6, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='18] it has been shown that even (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6) 0 < inf j∈N µj �µj ≤ sup j∈N µj �µj < +∞, which clearly implies M≈� M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We write M ∼= N if (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6) holds for the corresponding se- quences of quotients µ, ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Growth properties for abstractly given weight functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let ω : [0, +∞) → [0, +∞), we introduce the following growth and regularity conditions (ω1) : ω(2t) = O(ω(t)) t → +∞, (ω3) : log(t) = o(ω(t)) t → +∞, (ω4) : ϕω : t �→ ω(et) is a convex function on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' These conditions are named after [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ω1), (ω3) and (ω4) are standard assumptions in the theory of ultradifferentiable functions defined by so-called Braun-Meise-Taylor weight functions ω, see [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We write that ω has (ω0) if ω is continuous, non-decreasing, ω(t) = 0 for all t ∈ [0, 1] (normalization) and limt→+∞ ω(t) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Finally let us put W0 := {ω : [0, ∞) → [0, ∞) : ω has (ω0), (ω3), (ω4)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' If M ∈ LC then ωM ∈ W0, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' [9, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1] and the citations there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' N-functions in the weight sequence setting 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Basic definitions and abstractly given N-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We revisit the basic definitions from [11, Chapter I, §1, 3] and [1, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Consider f : [0, +∞) → [0, +∞) with the following properties: (I) f is right-continuous and non-decreasing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (II) f(0) = 0 and f(t) > 0 for all t > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (III) limt→+∞ f(t) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then we give the following definition, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' [11, Chapter I, §1, 3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let f : [0, +∞) → [0, +∞) be satisfying (I), (II) and (III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The function F : R → [0, +∞) defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1) F(x) := � |x| 0 f(t)dt, is called an N-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Every N-function F satisfies the following properties, see [11, Chapter I, §1, 4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 7]: (∗) F(0) = 0 (normalization) and F(x) > 0 for all x ̸= 0, (∗) F is even, non-decreasing, continuous, and convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) The convexity and F(0) = 0 imply that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2) ∀ 0 ≤ t ≤ 1 ∀ u ≥ 0 : F(tu) ≤ tF(u), see [11, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='14)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' This holds since by convexity we have F(tx+(1−t)y) ≤ tF(x)+(1−t)F(y) for all 0 ≤ t ≤ 1 and x, y ≥ 0 and then set y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 6 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' SCHINDL (∗) Finally, let us recall (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3) lim t→0 F(t) t = 0, lim t→+∞ F(t) t = +∞, see [11, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='15), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='16)], and which follows from (II) resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (III) for f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' When given two N-functions (or even arbitrary functions) F1, F2 : [0, +∞) → [0, +∞), then write F1 ⪯c F2 if (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4) ∃ K > 0 ∃ t0 > 0 ∀ t ≥ t0 : F1(t) ≤ F2(Kt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' If either F1 or F2 is non-decreasing then w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' we can restrict to K ∈ N>0 and this relation is clearly reflexive and transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In [11], F1 and F2 are called comparable, if either F1⪯cF2 or F2⪯cF1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' For this relation we are gathering several equivalent reformulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let F1, F2 : [0, +∞) → [0, +∞) be non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Assume that either F1 or F2 is normalized, convex and tending to infinity as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then the following are equivalent: (i) F1⪯cF2 holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) We have that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5) ∃ K, K1 ≥ 1 ∃ t0 > 0 ∀ t ≥ t0 : F1(t) ≤ K1F2(Kt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iii) We have that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6) ∃ C > 0 ∃ K ≥ 1 ∀ t ≥ 0 : F1(t) ≤ F2(Kt) + C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iv) We have that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7) ∃ K, K1 ≥ 1 ∃ C > 0 ∀ t ≥ 0 : F1(t) ≤ K1F2(Kt) + C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In particular, the above characterization applies if both F1 and F2 are N-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (i) ⇒ (ii) is trivial and (ii) ⇒ (i) follows by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2): If F2 is normalized and convex, then when given K1 > 1 we put t := K−1 1 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2) and hence K1F2(u) ≤ F2(K1u) for all u ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Similarly, if F1 is normalized and convex, then the assumption gives K−1 1 F1(t) ≤ F2(Kt) and so F1(t) ≤ K−1 1 F1(K1t) ≤ F2(KK1t) for all t ≥ t0 holds true which shows F1⪯cF2 when choosing K2 := KK1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (i) ⇒ (iii) is clear, since F2(t) ≥ 0 and F1 is non-decreasing take e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' C := F1(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iii) ⇒ (ii) When given C ≥ 1 then we have F2(Kt) + C ≤ 2F2(Kt) for all sufficiently large t if limt→+∞ F2(t) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Thus (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5) is verified with K1 := 2 and the same K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' If limt→+∞ F1(t) = +∞, then by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6) also limt→+∞ F2(t) = +∞ and the rest follows as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iii) ⇒ (iv) is trivial and (iv) ⇒ (iii) holds as (ii) ⇒ (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ This motivates the following definition, see [11, Chapter I, §3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We call two functions F1 and F2 equivalent, written F1 ∼c F2, if F1⪯cF2 and F2⪯cF1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In particular, for any N-function F we have that all Fk : t �→ F(kt), k > 0, are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In [11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2] it has been shown that F1∼cF2 if and only if L∗ F1 = L∗ F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We comment on relation ∼c for given N-functions F1, F2 and their corresponding right-derivatives f1, f2 appearing in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1): ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS 7 (i) On [11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 15] it is mentioned that if (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8) ∃ b ∈ (0, +∞) : lim t→+∞ F1(t) F2(t) = b, then F1∼cF2 holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Indeed, this implication holds for any non-decreasing functions F1, F2 : [0, +∞) → [0, +∞) such that either F1 or F2 is assumed to be convex and normal- ized: For any 0 < a ≤ 1 we clearly have aF2(u) ≤ F2(u) and if a > 1, then as in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2 the estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2) applied to t := a−1 gives aF2(u) ≤ F2(au) for all u ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The proof for F1 is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In particular, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8) holds true (with b = 1) if F1(t) = F2(t) for all t large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) Moreover, if limt→+∞ Fi(t) = +∞, i = 1, 2, then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8) holds with b = 1 if (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='9) ∃ C, D ≥ 1 ∀ t ≥ 0 : F1(t) − C ≤ F2(t) ≤ F1(t) + D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iii) In [11, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1] it has been shown that f1⪯cf2 implies F1⪯cF2 and in [11, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2] that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='10) ∃ b ∈ (0, +∞) : lim t→+∞ f1(t) f2(t) = b implies F1∼cF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In the theory of Orlicz classes also the following slightly different relation between functions is considered: (∗) Let F1, F2 be N-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In [11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1], see also [1, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3], it has been shown that LF1(Ω) ⊆ LF2(Ω) (as sets) if and only if (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11) ∃ t0 > 0 ∃ K ≥ 1 ∀ t ≥ t0 : F2(t) ≤ KF1(t), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' F2(t) = O(F1(t)) as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The sufficiency of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11) for having the inclusion LF1(Ω) ⊆ LF2(Ω) is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) Given F1, F2 : [0, +∞) → [0, +∞) we write F1 ⪯ F2 if (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11) holds true and F1 ∼ F2 if F1 ⪯ F2 and F2 ⪯ F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Note that this relation ∼ is precisely [11, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6)] and it is also the crucial one for the characterization of inclusions (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' equalities) of classes in the ultradifferentiable weight function setting, see [15, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) In view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2), for given N-functions F1, F2 we have that F1 ⪯ F2 implies F2⪯cF1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' For this implication one only requires that either F1 or F2 is normalized and convex, see the proof of (ii) ⇒ (i) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Consequently, if N-functions F1 and F2 are related by F1 ∼ F2, then they are also equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' From weight sequences to associated N-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M ∈ LC be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' When rewrit- ing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3) we obtain for all t ≥ 0 (note that µ1 ≥ 1): (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='12) ϕωM (t) = ωM(et) = � et µ1 ΣM(s) s ds = � t log(µ1) ΣM(eu) s sdu = � t log(µ1) ΣM(eu)du = � t 0 ΣM(eu)du, since ΣM(eu) = 0 for 0 ≤ u < log(µ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' This formula should be compared with [11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='10)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' ΣM ◦ exp is right-continuous, non-decreasing and clearly limt→+∞ ΣM(et) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Recall that ωM ∈ W0 and so ϕωM is convex, t �→ ϕωM (t) t is non-decreasing with limt→+∞ ϕωM (t) t = +∞ and finally ϕωM (0) = 0 is valid, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' [2, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5] and also [11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 8 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' SCHINDL Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' However, requirement (II) cannot be achieved for ΣM ◦ exp for any M ∈ LC: If M1 = M0(= 1), and so µ1 = 1, then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2) yields ΣM(e0) = ΣM(µ1) ≥ 1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' If M1 > M0 ⇔ µ1 > 1, then ΣM(et) = 0 for all 0 ≤ t < log(µ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Finally remark that, if M is log-convex with limj→+∞(Mj)1/j = +∞ but such that normalization fails, then 0 < µ1 < 1 and so ΣM(et) ≥ 1 for any t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Thus also in this case the first requirement in (II) is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' This failure is related to the fact that the first property in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3) and ϕωM (t) t > 0 for all t > 0 are not satisfied automatically for ϕωM , see the proofs and arguments in [11, Chapter I, §1, 5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 8-9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Thus ϕωM is formally not an N-function according to Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In order to overcome this technical problem we recall the following notion, see [11, Chapter I, §3, 3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 16]: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' A convex function Q is called the principal part of an N-function F if ∃ t0 > 0 ∀ t ≥ t0 : Q(t) = F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We have the following result, see [11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3] and the proof there: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let Q : [0, +∞) → [0, +∞) be a convex function such that limt→+∞ Q(t) t = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then there exists an N-function F such that Q is the principal part of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' More precisely, we even get that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='13) ∃ t0 > 0 ∀ t ≥ t0 : f(t) = q(t), with f denoting the function appearing in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1) of F and q denoting the non-decreasing and right- continuous function appearing in the representation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='14) Q(t) = � t a q(s)ds, see [11, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='10)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Here a ≥ 0 is such that Q(a) = 0 and we have t0 > a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let Q be a convex function such that limt→+∞ Q(t) t = +∞ and let F be the N-function according to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then we get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='15) ∃ C, D ≥ 1 ∀ t ≥ 0 : Q(t) − C ≤ F(t) ≤ Q(t) + D, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' This relation implies limt→+∞ F (t) Q(t) = 1 and so both F∼cQ and F ∼ Q holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' More generally, when given functions F1, F2 : [0, +∞) → [0, +∞) are satisfying F1(t) = F2(t) for all t large then we have F1(t) ≤ F2(t) + C, F2(t) ≤ F1(t) + D for all t ≥ 0 with C := max{F1(t) : 0 ≤ t ≤ t0} and D := max{F2(t) : 0 ≤ t ≤ t0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Hence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='15) follows by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8 (recall Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Note that Q is convex but normalization for Q (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' a = 0 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='14)) is not guaranteed in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Conversely, each convex function Q : [0, +∞) → [0, +∞) admitting the represen- tation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='14) for a non-decreasing and right-continuous function q with q(t) → +∞ satisfies also limt→+∞ Q(t) t = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' This holds since for all t ≥ a (see the proof of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='16) in [11, Chapter I, §1,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 7]): Q(2t) = � 2t a q(s)ds ≥ � 2t t q(s)ds ≥ q(t)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In particular, when applying these results to Q = ϕωM we get the following consequence: ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS 9 Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M ∈ LC be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then there exists an N-function FM such that ϕωM is the principal part of FM and so (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='16) ∃ C, D ≥ 1 ∀ t ≥ 0 : ϕωM (t) − C ≤ FM(t) ≤ ϕωM (t) + D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' This implies ϕωM ∼ FM and hence also ϕωM ∼cFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Moreover, if fM denotes the function appearing in the representation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1) of FM, then we even get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='17) ∃ t0 > 0 ∀ t ≥ t0 : fM(t) = ΣM(et).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In view of this equality we call ΣM ◦ exp the principal part of fM (see [11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We can apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8 to Q ≡ ϕωM because limt→+∞ ϕωM (t) t = lims→+∞ ωM(s) log(s) = +∞ by (ω3) (recall [9, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1] and the citations there).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In fact (ω3) for ωM is precisely [11, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6)] for Q = ϕωM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='16) follows by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='9, and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='17) holds by taking into account the representation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Note that by normalization of M we get ϕωM (0) = ωM(1) = 0 and so in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='14) we have a = 0 and q = ΣM ◦ exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ The following is an immediate consequence of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='16): Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M, L ∈ LC be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then FM⪯cFL if and only if ϕωM ⪯cϕωL and FM ⪯ FL if and only if ϕωM ⪯ ϕωL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M ∈ LC be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then the N-function FM from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11 is called the associated N-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We close this section by commenting on the relation between ϕωM and other notions of defining functions in the Orlicz setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' As seen above, for any given M ∈ LC we cannot expect that ϕωM is formally an N-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' On the other hand ϕωM can be used to illustrate the differences between appearing definitions for Orlicz classes in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In [14] an exhaustive study is provided and the different notions and conditions for the defining functions are compared, see also the literature citations there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) ϕωM coincides with an N-function (with FM) for sufficiently large values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) For any M ∈ LC the function ϕωM is always a Young function, see [14, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4]: ϕωM is convex, satisfies ϕωM (0) = ωM(1) = 0 by normalization and ϕωM (t) → +∞ as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) ϕωM is a strong Young function (see [14, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7]) if and only if µ1 = 1: Continuity is clear and ϕωM (t) > 0 for all t > 0 follows if and only if ΣM(et) > 0 for all t ≥ 0, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' This is clearly equivalent to µ1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) Finally, ϕωM is always an Orlicz function (see [14, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='9]), since ϕωM is never identically zero or infinity (follows again by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='12)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) Consequently, ϕωM provides (counter)-examples for the first two strict implications in [14, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7], see [14, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8]: Each N-function is a strong Young function and each strong Young function is an Orlicz function but each implication cannot be reversed in general;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' one can take M ∈ LC with µ1 = 1 for the first and M ∈ LC with µ1 > 1 for the second part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 10 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' SCHINDL 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Comparison between associated N-functions The goal of this Section is to give a connection resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' comparison between the growth relation ⪯ for weight sequences, crucially appearing in the theory of ultradifferentiable and ultraholomorphic functions, and the previously defined relations ⪯c and ⪯ for (associated) N-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Main statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We start with some immediate (known) observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' First, we have the following result providing a complete characterization for the relation ⪯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M, L ∈ LC be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then the following are equivalent: (i) The associated N-functions satisfy FM ⪯ FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) The functions ϕωM and ϕωL satisfy ϕωM ⪯ ϕωL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iii) The sequences M and L are related by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1) ∃ A ≥ 1 ∃ c ∈ N>0 ∀ j ∈ N : Mj ≤ A(Lcj)1/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (i) ⇔ (ii) holds by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) ⇔ (iii) First note that ϕωM ⪯ ϕωL precisely means ∃ K, D ≥ 1 ∀ t ≥ 0 : ωL(et) = ϕωL(t) ≤ KϕωM(t) + D = KωM(et) + D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Since ωM(t) = ωL(t) = 0 for all 0 ≤ t ≤ 1 by normalization of M and L we get ωL(t) ≤ KωM(t)+D for all t ≥ 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' ωL(t) = O(ωM(t)) and so ωM ⪯ ωL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Similarly, ωM ⪯ ωL implies ϕωM ⪯ ϕωL as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Consequently, the desired equivalence (ii) ⇔ (iii) follows by the first part in [4, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ Now let us proceed with relation ⪯c and gather some immediate consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M, L ∈ LC be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (i) If L ≤ M, then by definition ωM(t) ≤ ωL(t) for all t ≥ 0 and so ϕωM (t) ≤ ϕωL(t) for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In view of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11 we get FM⪯cFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) More generally, if L⪯M, then by definition and since M0 = N0 = 1 we have ωM(t) ≤ ωL(ht) for some h > 1 and all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Hence ∃ sh > 0 ∀ s ≥ sh : ϕωM (s) = ωM(es) ≤ ωL(es+log(h)) = ϕωL(s + log(h)) ≤ ϕωL(sh), since ϕωL is non-decreasing and s + log(h) ≤ sh ⇔ log(h) ≤ s(h − 1) for all s large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' This verifies ϕωM ⪯cϕωL and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11 implies again FM⪯cFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iii) Consequently, equivalent weight sequences yield equivalent associated N-functions and, in particular, this applies to the situation described in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iv) However, the converse implication in (iii) is not true in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' For this recall that by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4) we get ∀ ℓ > 0 ∀ t ≥ 0 : ϕωMℓ(t) = ωMℓ(et) = ℓωM(et/ℓ) = ℓϕωM(t/ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then, by following the arguments in (i) in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4, we see that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2) ∀ ℓ > 0 : ϕωM ∼cϕωMℓ , hence by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11 also FM∼cFMℓ for any ℓ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' However, for ℓ ̸= 1 the sequences M and M ℓ are not equivalent since limj→+∞(Mj)1/j = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (v) In particular, (iv) applies to the Gevrey-sequence M ≡ Gs, s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' It is known that ωGs ∼ t �→ t1/s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' ϕωGs ∼ t �→ et/s (see the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' So (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2) is verified but clearly Gs is not equivalent to Gs′ if s ̸= s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS 11 The aim of the next result is to characterize FM⪯cFL in terms of a growth relation between M and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M, L ∈ LC be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Consider the following assertions: (i) L⪯M is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) The associated N-functions FM and FL (see Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11) satisfy FM⪯cFL, equivalently ϕωM ⪯cϕωL is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iii) The sequences M and L satisfy (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3) ∃ c ∈ N>0 ∃ A ≥ 1 ∀ j ∈ N : Lj ≤ AMcj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then (i) ⇒ (ii) ⇒ (iii) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' If either M or L has in addition (mg), then also (iii) ⇒ (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' By (iv) in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2 the implication (i) ⇒ (ii) cannot be reversed in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (i) ⇒ (ii) This is shown in (i), (ii) in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) ⇒ (iii) By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2 relation FM⪯cFL is equivalent to ∃ K ≥ 1 ∃ C ≥ 1 ∀ t ≥ 0 : FM(t) ≤ FL(Kt) + C, and so by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11 (take w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' K ∈ N>0) ∃ K ∈ N>0 ∃ D ≥ 1 ∀ t ≥ 0 : ωM(et) = ϕωM (t) ≤ ϕωL(Kt) + D = ωL(etK) + D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We set s := et and hence this is equivalent to ∃ K ∈ N>0 ∃ D ≥ 1 ∀ s ≥ 1 : ωM(s) ≤ ωL(sK) + D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Recall that M, L ∈ LC implies (by normalization) ωM(s) = ωL(s) = 0 for all s ∈ [0, 1] and so the previous estimate holds true for any s ≥ 0 (with the same constants).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Thus (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5) yields for all j ∈ N: MKj = sup t≥0 tKj exp(ωM(t)) ≥ 1 eD sup t≥0 tKj exp(ωL(tK)) = 1 eD sup s≥0 sj exp(ωL(s)) = 1 eD Lj, and we are done when taking c := K and A := eD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iii) ⇒ (ii) Assume that (mg) holds for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' By assumption (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3) and an iterated application of (mg) we get (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4) ∃ c ∈ N>0 ∃ A, B ≥ 1 ∀ k ∈ N : Lk ≤ AMck ≤ ABkM c k, thus by definition of associated weight functions ωMc(t) ≤ ωL(Bt) + log(A) for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Conse- quently, since ϕωL(t) → +∞ and by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2) we get for all t large enough ϕωMc(t) ≤ ϕωL(t + log(B)) + log(A) ≤ ϕωL(2t) + log(A) ≤ 2ϕωL(2t) ≤ ϕωL(4t), hence ϕωMc⪯cϕωL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2) we have that ϕωMc∼cϕωM and so ϕωM ⪯cϕωL is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11 yields the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' If L has in addition (mg), then by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3) and iterating (mg) first we get ∃ c ∈ N>0 ∃ A, B1 ≥ 1 ∀ j ∈ N : Lcj ≤ Bcj 1 Lc j ≤ AcBcj 1 M c cj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 12 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' SCHINDL Thus (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4) is verified for all k ∈ N with k = cj, j ∈ N arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' For the remaining cases let k with cj < k < cj + c for some j ∈ N and then, since both M and L are also non-decreasing, we get for some C ≥ 1 Lk ≤ Lcj+c ≤ Cc(j+1)LcLcj ≤ Cc(j+1)LcAcBcj 1 M c cj ≤ (AC)cLc(B1C)kM c k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Summarizing, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4) is verified for all k ∈ N and the rest follows as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ Thus we have the following characterization: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M, L ∈ LC be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Assume that either M or L has in addition (mg), then the following are equivalent: (i) The associated N-functions FM and FL are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) The functions ϕωM and ϕωL are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iii) The sequences M and L satisfy ∃ c, d ∈ N>0 ∃ A, B ≥ 1 ∀ j ∈ N : Mj ≤ ALcj, Lj ≤ BMdj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We close this section by comparing the characterizing conditions for M and L in the previous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M, L ∈ LC be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) We have Lj ≥ 1 for all j ∈ N and so (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1) implies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3) (with M, L interchanged).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) On the other hand, recall that by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2) we get that FM ⪯ FL implies FL⪯cFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Summariz- ing, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1) ⇐⇒ FM ⪯ FL =⇒ FL⪯cFM =⇒ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3), and the last implication can be reversed provided that either M or L has in addition (mg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' On sufficiency conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We want to find sufficient conditions for given sequences M, L in order to ensure FM⪯cFL resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' FM∼cFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In explicit applications and for constructing weight sequences M it is often convenient to start with the corresponding sequence of quotients µ = (µj)j∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Note that when involving µ we get automatically growth conditions for the counting function ΣM as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M, L ∈ LC be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (a) The following assertions are equivalent: (i) We have that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6) ∃ A ≥ 1 ∃ k > 0 ∃ t0 > 0 ∀ t ≥ t0 : ΣM(t) ≤ AΣL(tk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) We have that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7) ∃ B ≥ 1 ∃ k > 0 ∃ j0 ∈ N>0 ∀ j ≥ j0 : λ⌈j/B⌉ ≤ µk j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (b) Any of the equivalent conditions listed in (a) implies that FM⪯cFL (equivalently ϕωM ⪯cϕωL) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The proof shows that in (a) we can take A = B and the same k and note that the result becomes trivial for M = L (set A = B = k = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (a)(i) ⇒ (ii) Let j0 ∈ N>0 be minimal to ensure µj0 ≥ t0 (note that limj→+∞ µj = +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' For any t ≥ µj0 we have µj ≤ t < µj+1 for some j ∈ N>0, j ≥ j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then by assumption j = ΣM(t) ≤ AΣL(tk) ⇔ ΣL(tk) ≥ j A and so tk ≥ λ⌈j/A⌉ (note that ΣL(tk) ∈ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' When taking t := µj we get (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7) with B := A and the same k > 0 for all j ≥ j0 with µj < µj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS 13 If j ≥ j0 with µj = µj+1, then µj = µj+1 = · · · = µj+d < µj+d+1 for some d ∈ N>0 and thus j + d = ΣM(t) for µj+d ≤ t < µj+d+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' This yields ΣL(tk) ≥ j+d A , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' tk ≥ λ⌈(j+d)/A⌉ ≥ λ⌈(j+i)/A⌉ for all 0 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Put t := µj+d(= · · · = µj) in order to verify (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7) with B := A and the same k > 0 for all j ≥ j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (a)(ii) ⇒ (i) Let t ≥ 0 be such that µj ≤ t < µj+1 for some j ≥ j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then µk j ≤ tk < µk j+1 and so tk ≥ µk j ≥ λ⌈j/B⌉ which gives ΣM(t) = j and ⌈j/B⌉ ≤ ΣL(tk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Thus (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6) follows when j ≤ Aj/B ≤ A⌈j/B⌉ is ensured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' So we can take A := B, the same k > 0 and t0 := µj0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (b) If (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6) is valid, then ΣM(es) ≤ AΣL(esk) for some A ≥ 1 and all sufficiently large s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Thus by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='17) we can find s0 > 0 such that fM(s) ≤ AfL(ks) for all s ≥ s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then FM⪯cFL holds similarly as shown in [11, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1] (with A = 1 there): For all s ≥ s0 we get FM(s) = � s 0 fM(u)du = � s0 0 fM(u)du + � s s0 fM(u)du ≤ � s0 0 fM(u)du + A � s s0 fL(ku)du ≤ � s0 0 fM(u)du + A � s 0 fL(ku)du = � s0 0 fM(u)du + A k � ks 0 fL(v)dv = FM(s0) + A k FL(ks), hence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7) between FM and FL follows when choosing C := FM(s0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2 yields the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ Another sufficiency criterion expressed in terms of the counting functions ΣM, ΣL is the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M, L ∈ LC be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Assume that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8) ∃ b ∈ (0, +∞) : lim t→+∞ ΣM(t) ΣL(t) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then we get FM∼cFL (equivalently ϕωM ∼cϕωL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Note: For M = L property (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8) holds trivially with b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='17) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8) we have limt→+∞ fM (t) fL(t) = b > 0 and so [11, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2] yields FM∼cFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ Let us now study condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8) in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M, L ∈ LC be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Assume that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='9) ∃ c, j0 ∈ N>0 ∀ j ≥ j0, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' µj < µj+1, ∃ dj ∈ N>0 : λcj ≤ µj < µj+1 ≤ λcj+dj, with dj ∈ N>0 satisfying limj→+∞ dj j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8) holds true (with b = 1 c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' For all t ≥ µj0 we find j ≥ j0 such that µj ≤ t < µj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then ΣM(t) = j and by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='9) we get cj ≤ ΣL(t) < cj + dj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Thus j cj + dj < ΣM(t) ΣL(t) ≤ j cj , ∀ µj ≤ t < µj+1, j ≥ j0, and since limj→+∞ dj j = 0 we get (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8) with b := 1 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ Note that it is enough to require the existence of a sequence dj for j such that µj < µj+1: If j ≥ j0 and µj = µj+1 = · · · = µj+ℓ < µj+ℓ+1 for some ℓ ∈ N>0, then by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='9) we get λcj ≤ λcj+cℓ ≤ µj = µj+1 = · · · = µj+ℓ < µj+ℓ+1 ≤ λcj+cℓ+dj+ℓ, 14 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' SCHINDL and so for t with µj+ℓ ≤ t < µj+ℓ+1 we have ΣM(t) = j + ℓ and cj + cℓ = c(j + ℓ) ≤ ΣL(t) < cj + cℓ + dj+ℓ = c(j + ℓ) + dj+ℓ yielding the same estimate as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (On the other hand, by switching to an equivalent sequence, we can assume µj < µj+1 for all j ∈ N>0, see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=') We prove now the following characterization for (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M, L ∈ LC be given such that µj < µj+1 for all j ∈ N>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then the following are equivalent: (i) We have that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='10) ∃ b ∈ (0, +∞) : lim t→+∞ ΣM(t) ΣL(t) = b, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) We have that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11) ∃ b ∈ (0, +∞) ∃ d ∈ N ∀ 0 < c1 < b < c2 ∃ j0 ∈ N>0 ∀ j ≥ j0 : λ⌈j/c2⌉−d ≤ µj < λ⌈j/c1⌉+d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iii) We have that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='12) ∃ b ∈ (0, +∞) ∀ 0 < c1 < b < c2 ∃ j0 ∈ N>0 ∀ j ≥ j0 ∃ dj ∈ N>0 : λ⌈j/c2⌉−dj ≤ µj < λ⌈j/c1⌉+dj, with dj ∈ N>0 satisfying limj→+∞ dj j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The proof shows that in all assertions the value b is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (i) ⇒ (ii) By assumption, ∀ ǫ > 0 ∃ tǫ > 0 ∀ t ≥ tǫ : ΣL(t)(b − ǫ) < ΣM(t) < (b + ǫ)ΣL(t), thus we find jǫ ∈ N>0 large satisfying µjǫ ≥ tǫ so that for all t with µj ≤ t < µj+1 for some j ≥ jǫ we get (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='13) ΣL(t)(b − ǫ) < j = ΣM(t) < (b + ǫ)ΣL(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The first estimate in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='13) yields ΣL(t) < j b−ǫ ≤ ⌈ j b−ǫ⌉ and since ΣL(t) ∈ N we have ΣL(t) ≤ ⌈ j b−ǫ⌉ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let now c1 < b be arbitrary but fixed and take ǫ1 small enough to ensure 1 b−ǫ1 ≤ 1 c1 ⇔ c1 ≤ b − ǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Note that the choice of ǫ1 is only depending on chosen c1 but not on j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Thus ΣL(t) < ⌈ j c1 ⌉ and so t < λ⌈ j c1 ⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We take t := µj and get µj < λ⌈ j c1 ⌉, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' the second half of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11) for all j ≥ jǫ1 (since µj < µj+1 for all j) with d := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Similarly, the second estimate in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='13) gives ⌊ j b+ǫ⌋ ≤ j b+ǫ < ΣL(t) and so t ≥ λ⌊ j b+ǫ ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We let c2 > b be arbitrary but fixed and choose ǫ2 > 0 small enough to ensure 1 b+ǫ2 ≥ 1 c2 ⇔ c2 ≥ b + ǫ2 and so ⌊ j b+ǫ2 ⌋ ≥ ⌈ j b+ǫ2 ⌉ − 1 ≥ ⌈ j c2 ⌉ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Take t := µj to get the first half of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11) for all j ≥ jǫ2 with d := 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Summarizing, we are done when taking j0 := max{jǫ1, jǫ2} and note that d can be taken uniformly and not depending on chosen c1, c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) ⇒ (iii) This is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iii) ⇒ (i) Let b ∈ (0, +∞) be given and take arbitrary (close) 0 < c1 < b < c2 but from now on fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let t ≥ µj0 and so µj ≤ t < µj+1 for some j ≥ j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='12) gives λ⌈j/c2⌉−dj ≤ µj ≤ t < µj+1 < λ⌈(j+1)/c1⌉+dj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS 15 So ΣM(t) = j and the first estimate implies ΣL(t) ≥ ⌈ j c2 ⌉ − dj ≥ j c2 − dj, whereas the last estimate yields ΣL(t) ≤ ⌈ j+1 c2 ⌉ + dj+1 − 1 ≤ j+1 c1 + dj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Summarizing, for all such t we obtain j j/c1 + 1/c1 + dj+1 ≤ ΣM(t) ΣL(t) ≤ j j/c2 − dj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then note that dj+1 j = dj+1 j+1 j+1 j → 0 as j → +∞ ⇔ t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Hence, as c1, c2 → b we get that ΣM(t) ΣL(t) → b as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' From N-functions to associated weight sequences The aim is to reverse the construction from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' we start with an abstractly given N- function G, associate to it a sequence M G and study then the relation between G and the associated N-function FMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let F be an N-function and first we introduce (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1) ωF (t) := F(log(t)), t ≥ 1, ωF (t) := 0, 0 ≤ t < 1 (normalization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' By the properties of F we have that ωF belongs to the class W0: Concerning (ω4) note that ϕωF (t) = ωF (et) = F(t) for all t ≥ 0, concerning (ω3) we remark that ωF (t) log(t) = ωF (es) s = F (s) s for all t > 1 ⇔ s > 0 and the last quotient tends to +∞ as s → +∞, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Note: When given ω ∈ W0, then one can put (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2) Fω(t) := ϕω(|t|) = ω(e|t|), t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Fω satisfies all properties to be formally an N-function (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1) except necessarily the first part in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3) and also Fω(t) > 0 for all t ̸= 0 is not clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Thus the set of all N-functions does not coincide with the class W0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' However, one can overcome this technical problem for Fω by passing to an equivalent (associated) N-function when taking into account Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8 analogously as it has been done before with ϕωM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We consider the so-called Legendre-Fenchel-Young-conjugate of ϕωF which is given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3) ϕ∗ ωF (s) := sup{|s|t − ϕωF (t) : t ≥ 0} = sup{|s|t − F(t) : t ≥ 0}, s ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' ϕωF is non-decreasing, convex by assumption, ϕωF (0) = ωF (e0) = F(0) = 0 and limt→+∞ ϕωF (t) t = lims→+∞ ωF (s) log(s) = +∞ as seen before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Thus we get, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' [2, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3]: ϕ∗ ωF is convex, ϕ∗ ωF (0) = 0, s �→ ϕ∗ ωF (s) s is non-decreasing and tending to +∞ as s → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Finally ϕ∗∗ ωF (s) = ϕωF (s) holds true for all s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Next let us introduce the associated sequence M F = (M F j )j by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4) M F j := sup t>0 tj exp(ωF (t)) = sup t≥1 tj exp(ωF (t)) = sup t≥1 tj exp(F(log(t))), j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The second equality holds by normalization and this formula should be compared with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Hence for any j ∈ N we get M F j = sup t>0 tj exp(ωF (t)) = exp sup t>0 (j log(t) − ωF(t)) = exp sup s∈R (js − ωF (es)) = exp sup s≥0 (js − ωF (es)) = exp sup s≥0 (js − ϕωF (s)) = exp(ϕ∗ ωF (j)), 16 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' SCHINDL which implies M F ∈ LC by the properties of the conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Note that by definition (normalization) one has ωF (es) = 0 for all −∞ < s ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Finally, by [18, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3] applied to ω = ωF , see also [15, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7] and [8, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5] with weight matrix parameter x = 1 there, we obtain (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5) ∃ C ≥ 1 ∀ t ≥ 0 : ωMF (t) ≤ ωF (t) ≤ 2ωMF (t) + C, hence by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6) ∃ C ≥ 1 ∀ t ≥ 0 : ϕωMF (t) ≤ ωF (et) = F(t) ≤ 2ϕωMF (t) + C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In order to avoid confusion let from now in this context G be the given N-function, M G the sequence defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4) and FMG the associated N-function from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11 (applied to the sequence M G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let G be an N-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then the sequence M G is called the associated weight sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let G be an N-functions and let FMG be the N-function associated with the sequence M G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then we get (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7) ∃ A, B ≥ 1 ∀ t ≥ 0 : FMG(t) ≤ G(t) + A ≤ 2FMG(t) + B ≤ FMG(2t) + B, and this implies both G ∼ FMG ∼ ϕωMG and G∼cFMG∼cϕωMG .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11 applied to M G and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6) yield ∃ C, C1 ≥ 1 ∀ t ≥ 0 : G(t) ≤ 2ϕωMG(t) + C ≤ 2FMG(t) + C + C1, and ∃ C2 ≥ 1 ∀ t ≥ 0 : FMG(t) ≤ ϕωMG (t) + C2 ≤ G(t) + C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' These estimates prove the first two parts in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7) and the last one there follows by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2), so by convexity and normalization of FMG, see also the estimate in (i) in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4 applied to a := 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The desired relations follow now by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7), Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ The next result gives the (expected) equivalence when starting with a weight sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let L ∈ LC be given and FL the associated N-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then we get ∃ A, B ≥ 1 ∀ j ∈ N : 1 ALj ≤ M FL j ≤ BLj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' This estimate implies M FL≈L, FL∼cFMFL and FL ∼ FMFL .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We apply the previous constructions to the associated N-function FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' For all t ≥ 1 we have ωFL(t) = FL(log(t)) and so by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='16) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8) ∃ C, D ≥ 1 ∀ t ≥ 1 : ωL(t)−C = ϕωL(log(t))−C ≤ ωFL(t) ≤ ϕωL(log(t))+D = ωL(t)+D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Because L ∈ LC we have ωL(t) = 0 for all t ∈ [0, 1] (normalization) and ωFL(t) = 0 holds for all t ∈ [0, 1] by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Thus (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8) is valid for any t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' By combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8) with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5) we arrive at ∃ C, D ≥ 1 ∀ j ∈ N : e−DLj ≤ M FL j ≤ eCLj, hence the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' This relation clearly implies M FL≈L and so, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3 (recall also Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2) the equiva- lence FL∼cFMFL .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Moreover, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1) is verified with c = 1 (pair-wise) and hence Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1 gives that FL ∼ FMFL as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS 17 We continue with the following observations: (∗) Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2 suggests that for abstractly given N-functions G it is important to have in- formation about FMG resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' ϕωMG and to study the associated weight sequence M G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In particular, in view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='12) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='17) the knowledge about the counting function ΣMG is useful and this amounts to study the sequence of quotients µG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) On the other hand, when desired growth behaviours are expressed in terms of µG, it is an advantage not to compute first M G via given G and then the corresponding quotient sequence µG but to come up with a property for G directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) This can be achieved by relating µG directly to G as follows: Put (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='9) vG(t) := exp(−G(log(t))), t ≥ 1, vG(t) := 1, 0 ≤ t < 1, and so (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4) takes alternatively the following form: M G j = sup t>0 tjvG(t), j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' This should be compared with [19, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 6], [17, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Indeed, M G coincides with the crucial associated sequence M vG in [19], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' By the assumptions on G we get that vG is a (normalized and convex) weight in the notion of [19], [17], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' in the setting of weighted spaces of entire functions, see also the literature citations in these papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) Let now (tj)j∈N be the sequence such that tj ≥ 0 is denoting a/the global maximum point of the mapping t �→ tjvG(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) The advantage when considering vG is that in [19, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5)] we have derived the following relation between the sequence of quotients µG (set µG 0 := 1) and (tj)j∈N: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='10) ∀ j ∈ N : tj ≤ µG j+1 ≤ tj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Note: For j = 0 we have put t0 := 1 and so equality with µG 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In fact for j = 0 we can choose any t ∈ [0, 1] as t0 since vG is non-increasing and normalized, and by the convention 00 := 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' So tj ≥ 1 for any j ∈ N because j �→ tj is non-decreasing and since limj→+∞ µG j = +∞ we also get tj → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' For concrete given G (belonging to C1) the concrete computation for the values of tj might be not too difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then put (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11) ΣG(t) := |{j ∈ N>0 : tj ≤ t}|, t ≥ 0, and ΣG : [0, +∞) → N is a right-continuous non-decreasing function with ΣG(t) = 0 for all 0 ≤ t < 1 and ΣG(t) → +∞ as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' By taking into account (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='10) and the definition of ΣMG and ΣG we get: Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let G be an N-function and M G the associated weight sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then the counting functions ΣG and ΣMG are related by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='12) ∀ t ≥ 0 : ΣMG(t) ≤ ΣG(t) + 1 ≤ ΣMG(t) + 1, which yields lim t→+∞ ΣMG(t) ΣG(t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Using ΣG we introduce (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='13) ϕG(x) := � |x| 0 ΣG(et)dt, x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 18 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' SCHINDL Note that, analogously to the explanations for ϕωM given in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6 in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2 we have that ϕG is formally not an N-function since tj ≥ 1 for all j ≥ 1 and so by definition ΣG(e0) ≥ 1 ̸= 0 if t1 = 1 or ΣG(et) = 0 for all 0 ≤ t < log(t1) if t1 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We summarize the whole information in the final statement of this section: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let G be an N-function, M G the associated weight sequence, FMG the associated N-function and ϕG given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then G∼cFMG∼cϕωMG ∼cϕG, G ∼ FMG ∼ ϕωMG ∼ ϕG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The first two parts are shown in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The last one follows by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4: We use (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='12) and the representations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='13) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='12) in order to get analogously as in the proof of (b) in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='14) ∃ C ≥ 1 ∀ t ≥ 0 : ϕG(t) ≤ ϕωMG(t) ≤ ϕG(t) + t ≤ 2ϕG(t) + C ≤ ϕG(2t) + C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The second last estimate in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='14) holds since limt→+∞ ϕG(t) t = +∞ and the last one by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2) applied to ϕG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Both requirements on ϕG are valid by the representation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='13), see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='14) implies both ϕωMG ∼cϕG and ϕωMG ∼ ϕG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5 shows that the whole information concerning growth and regularity properties of G is already expressed by involving a certain associated weight sequence M G and its related/associated N-function FMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' On complementary N-functions and dual sequences We give a connection between the so-called complementary N-functions F c and the dual sequences D in the weight sequence setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Note that in the theory of N-functions one naturally has the pair (F, F c) and thus also (FM, F c M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Similarly for each M ∈ LC one can naturally assign the dual sequence D ∈ LC and we show that F c M is closely related to D via an integral representation formula using the counting-function log ◦ΣD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Complementary N-functions in the weight sequence setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let F be an N-function given by the representation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1) with right-derivative f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' First, introduce (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1) f c(s) := sup{t ≥ 0 : f(t) ≤ s}, s ≥ 0, thus f c is the right-inverse of f, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' it is the right-inverse of the right-derivative of F, see [11, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' It is known that f c satisfies (I) − (III) in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The complementary N-function F c, see [11, Chapter I, §2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 11], is defined by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2) F c(x) := � |x| 0 f c(t)dt, x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In [11, Chapter I, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='9), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 13] it is mentioned that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3) F c(s) = max t≥0 {|s|t − F(t)}, s ∈ R, and this formula can be considered as an equivalent definition for F c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We comment on the comparison of growth relations ⪯c and ⪯ between (associated) N-functions F, G and their complementary N-functions F c, Gc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (i) In [11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2] it has been shown that F⪯cG yields Gc⪯cF c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In particular, if two N-functions are equivalent then their complementary N-functions as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS 19 (ii) If F ⪯ G, then G(t) ≤ CF(t) + D for some C, D ≥ 1 and all t ≥ 0 and so (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3) gives for any s ∈ R Gc(s) = max t≥0 {|s|t − G(t)} ≥ max t≥0 {|s|t − CF(t)} − D = C max t≥0 {C−1|s|t − F(t)} − D = CF c(s/C) − D, see also the proof of [15, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' This relation implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7) and so Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2 yields that F c⪯cGc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' When one can choose C = 1, then also Gc ⪯ F c follows but in general this implication is not clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' By combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3) we get ϕ∗ ωF = F c, and since M F j = exp(ϕ∗ ωF (j)), for this see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4) and the computations below this equation, it follows that ∀ j ∈ N : M F j = exp(F c(j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let now M ∈ LC be given, then write F c M and f c M for the functions considered before w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' to the associated N-function FM and the corresponding right-derivative fM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Moreover, in view of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1), let us introduce (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4) ΓM(s) := sup{t ≥ 0 : ΣM(et) ≤ s}, s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Finally we set (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5) ϕc ωM (x) := � |x| 0 ΓM(s)ds, x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='17) it follows that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6) ∃ s0 > 0 ∀ s ≥ s0 : f c M(s) = ΓM(s), because both functions are non-decreasing and fM(t) = ΣM(et) for all t ≥ t0, with t0 the value appearing in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' So (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6) is valid for all s ≥ s0 := fM(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Using this identity we can prove the analogous result of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11 for the functions F c M and ϕc ωM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M ∈ LC be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then we get ∃ C, D ≥ 1 ∀ t ≥ 0 : ϕc ωM (t) − C ≤ F c M(t) ≤ ϕc ωM (t) + D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' This implies that both ϕc ωM ∼cF c M and ϕc ωM ∼ F c M hold true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We use (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6) and the representations (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5) (recall also the arguments in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' For all s ≥ s0 (with s0 from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6)) we get ϕc ωM (s) = � s 0 ΓM(t)dt = � s0 0 ΓM(t)dt + � s s0 ΓM(t)dt = � s0 0 ΓM(t)dt + � s s0 f c M(t)dt ≤ � s0 0 ΓM(t)dt + � s 0 fM(t)dt = ϕc ωM (s0) + F c M(s), hence ϕc ωM (s) ≤ F c M(s) + C for all s ≥ 0 when choosing C := ϕc ωM (s0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Similarly, for all s ≥ s0 F c M(s) = � s 0 f c M(t)dt = � s0 0 f c M(t)dt + � s s0 f c M(t)dt = � s0 0 f c M(t)dt + � s s0 ΓM(t)dt ≤ � s0 0 f c M(t)dt + � s 0 ΓM(t)dt = F c M(s0) + ϕc ωM (s), 20 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' SCHINDL hence F c M(s) ≤ ϕc ωM (s) + D for all s ≥ 0 when choosing D := F c M(s0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ On the other hand, formula (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3) applied to ϕωM yields (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7) sup t≥0 {|s|t − ϕωM (t)} = ϕ∗ ωM (s), s ∈ R, the Legendre-Fenchel-Young-conjugate of ϕωM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' By combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='16), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3) applied to FM and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7) we get (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8) ∃ C, D ≥ 1 ∀ s ∈ R : −C + F c M(s) ≤ ϕ∗ ωM (s) ≤ F c M(s) + D, see the estimate in (ii) in Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Hence F c M∼cϕ∗ ωM and F c M ∼ ϕ∗ ωM hold true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' When combining this with Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3 we arrive at the following result: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M ∈ LC be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then ϕc ωM ∼cF c M∼cϕ∗ ωM , ϕc ωM ∼ F c M ∼ ϕ∗ ωM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Finally, we apply Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4 to the associated weight sequence M G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let G be an N-function, M G ∈ LC the associated sequence defined via (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4) and ϕG given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then ϕ∗ ωMG ∼cϕc ωMG∼cF c MG∼cGc∼c(ϕG)c, ϕ∗ ωMG ∼ ϕc ωMG ∼ F c MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Concerning ∼c, the first and the second equivalence hold by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4, the third and the fourth one by taking into account (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7), Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5 (see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='14)) and (i) in Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Here (ϕG)c is given in terms of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' For ∼ we use the same results, however F c MG ∼ Gc and Gc ∼ (ϕG)c are not clear in general: In order to conclude we want to apply (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3) and so in the relations only an additive constant should appear, see (ii) in Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' However, in both (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='14) we also have the multiplicative constant 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Complementary associated N-functions versus dual sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In this section the aim is to see that ΓM in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4) and crucially appearing in the representation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5) is closely connected to the counting function ΣD, with D denoting the so-called dual sequence of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Moreover, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4) should be compared with the formula for the so-called bidual sequence of M in [5, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='41, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' For given M ∈ LC we introduce the dual sequence D = (Dj)j defined in terms of the corresponding quotient sequence δ as follows, see [5, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='40, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 81]: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='9) ∀ j ≥ µ1(≥ 1) : δj+1 := ΣM(j), δj+1 := 1 ∀ j ∈ Z, −1 ≤ j < µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then we set Dj := j� k=0 δk, j ∈ N, hence D ∈ LC with 1 = D0 = D1 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Recall that by [5, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='27] the function νm in [5] precisely denotes the counting function ΣM (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2)) and note that concerning the sequence of quotients there exists an index-shift: more precisely we have mj ≡ µj+1 for all j ∈ N with m = (mj)j used in [5] and [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Moreover, a weight sequence in [5] means a sequence satisfying all requirements from class LC except M0 ≤ M1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' see [5, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M ∈ LC be given, we analyze now ΓM: ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS 21 (∗) Obviously, lims→+∞ ΓM(s) = +∞ and since limj→+∞ µj = +∞ we have µj < µj+1 for infinitely many indices j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) Then recall that ΣM(et) ∈ N and that ΣM(et) = 0 for all 0 ≤ t < log(µ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' By normalization we get log(µj) ≥ log(µ1) ≥ log(1) = 0 for all j ∈ N>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) Case I: Assume that 1 = µ1 = · · · = µd < µd+1 for some d ∈ N>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then et ≥ µd = 1 and so ΣM(et) ≥ d for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Hence the set of values t ≥ 0 satisfying ΣM(et) ≤ s in the definition of ΓM(s) is empty for all values s with 0 ≤ s < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In this case we put ΓM(s) := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Note that d is finite because limj→+∞ µj = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let now s be such that j ≤ s < j + 1 for some j ∈ N>0 with j ≥ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In view of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2 in general it is not clear that we can find t ≥ 0 such that ΣM(et) = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In fact this holds true if and only if µj < µj+1 because for all t ≥ 0 with log(µj) ≤ t < log(µj+1) we get ΣM(et) = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We have Σ(elog(µj+1)) ≥ j + 1 > s and consequently for all such indices j, in particular for j = d, we obtain ΓM(s) = log(µj+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' If j ≥ d is such that µj = µj+1, then j ≥ d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Thus there exist ℓ, c ∈ N>0 with d ≤ ℓ ≤ j − 1 and such that µℓ < µℓ+1 = µℓ+2 = · · · = µℓ+c = µj = µj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' For all t with log(µℓ) ≤ t < log(µℓ+1) we get ΣM(et) = ℓ < j and since ΣM(elog(µℓ+1)) = ΣM(elog(µj+1)) ≥ j + 1 > s we have again ΓM(s) = log(µj+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) Case II: Assume that µ1 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' First we have ��M(s) = log(µ1)(> 0) for all 0 ≤ s < 1 since ΣM(et) = 0 for all 0 ≤ t < log(µ1) and ΣM(elog(µ1)) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let now j ≤ s < j + 1 for some j ∈ N>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Similarly as above, if µj < µj+1 then we get ΓM(s) = log(µj+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' If µj = µj+1, then we distinguish: Either there exist ℓ, c ∈ N>0 with 1 ≤ ℓ ≤ j − 1 such that µℓ < µℓ+1 = µℓ+2 = · · · = µℓ+c = µj = µj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then again ΓM(s) = log(µj+1) by the same reasons as in Case I before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Finally, if this choice is not possible, then this precisely means µ1 = · · · = µj = µj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Since µ1 > 1 we have ΣM(et) = 0 for all 0 ≤ t < log(µ1) and ΣM(et) ≥ j + 1 > s for all t ≥ log(µj+1) = log(µ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Thus ΓM(s) = log(µj+1) = log(µ1) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Summarizing, we have shown the following statement: Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M ∈ LC be given with quotient sequence (µj)j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (a) If 1 = µ1 = · · · = µd < µd+1 for some d ∈ N>0, then ΓM(s) = 0, ∀ 0 ≤ s < d, ΓM(s) = log(µj+1), ∀ j ≤ s < j + 1, ∀ j ≥ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (b) If µ1 > 1, then ΓM(s) = log(µj+1), ∀ j ≤ s < j + 1, ∀ j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Next let us study ΣD in detail, see also the proof of [5, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='42]: (∗) Let t ≥ 0, then the value ΣD(t) is equal to the maximal integer j ∈ N>0 satisfying δj ≤ t if it exists and otherwise equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' By definition in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='9) we have δj ∈ N>0 for all j ∈ N and so ΣD(t) = 0 for 0 ≤ t < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) In fact δj = 1 for all j ∈ N>0 satisfying j < µ1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The largest of those integers j coincides with ⌊µ1 + 1⌋ ≥ 2 if µ1 /∈ N>0 (in this case µ1 > 1) and with µ1 if µ1 ∈ N>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Moreover, δj = 1 for all j satisfying µ1 + 1 ≤ j < µ2 + 1 if such an integer j exists (case I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In particular, this holds if µ2 ≥ µ1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' If not, then δj = d ∈ N≥2 for the minimal integer j satisfying j ≥ µ1 + 1 (case II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' This occurs if for this minimal integer already j ≥ µd + 1 is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Note that d is finite since 22 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' SCHINDL limj→+∞ µj = +∞ and let d be such that µd + 1 ≤ j < µd+1 + 1 for j minimal satisfying j ≥ µ1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) Consider 1 ≤ t < 2 and distinguish: In the first case ΣD(t) is the largest integer j satisfying µ1 + 1 ≤ j < µ2 + 1 and in the second case ΣD(t) coincides with the largest integer j satisfying j < µ1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Since µ1 + 1 ≥ 2 in both cases the existence of such an integer j is ensured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) More generally, in case I if n ≤ t < n + 1 with n ∈ N>0, then ΣD(t) is the largest integer j satisfying j < µn+1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) In case II, if n ≤ t < n + 1 for some n ∈ N>0 with n + 1 ≤ d, then ΣD(t) is (still) the largest integer j such that j < µ1 + 1 since for the minimal integer j ≥ µ1 + 1 we have δj = ΣM(j − 1) ≥ ΣM(µd) = d > t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The last equality holds since µd < µd+1 by the choice of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' If n ≤ t < n + 1 for some n ∈ N>0 with n ≥ d, then δj coincides with largest integer j satisfying j < µn+1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' For n = d recall that at least one integer j exists with µd + 1 ≤ j < µd+1 + 1 and so δj = ΣM(j − 1) = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Summarizing, we have shown the following statement (see also [5, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='42]): Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M ∈ LC be given and D its dual sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then ΣD(t) = 0, ∀ 0 ≤ t < 1, and: (a) If for the minimal integer j satisfying j ≥ µ1 + 1 one has µd + 1 ≤ j < µd+1 + 1 for some d ∈ N>0, d ≥ 2, then ΣD(t) = max{j ∈ N>0 : j < µ1 + 1}, ∀ n ∈ N>0, n < d, ∀ n ≤ t < n + 1, ΣD(t) = max{j ∈ N>0 : j < µn+1 + 1}, ∀ n ∈ N>0, n ≥ d, ∀ n ≤ t < n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (b) If for the minimal integer j satisfying j ≥ µ1 + 1 one has µ1 + 1 ≤ j < µ2 + 1, then ΣD(t) = max{j ∈ N>0 : j < µn+1 + 1}, ∀ n ∈ N>0, ∀ n ≤ t < n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' When combining Propositions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7 we are able to establish a connection between the counting functions ΓM and ΣD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M ∈ LC be given and D its dual sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='10) ∃ t0 > 0 ∀ t ≥ t0 : ΓM(t) ≤ log(ΣD(t)) ≤ ΓM(t) + 1, and this estimate implies lim t→+∞ ΓM(t) log(ΣD(t)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='10) we can take t0 := d ∈ N>0 such that for the minimal integer j ≥ µ1 + 1 we get µd + 1 ≤ j < µd+1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' First, by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7 we get that ΣD(t) coincides with the maximal integer j < µn+1+1 for all t ≥ 0 satisfying n ≤ t < n + 1 and all n ≥ d with d ∈ N>0 such that for the minimal integer j ≥ µ1 + 1 we get µd + 1 ≤ j < µd+1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In particular, for all such n we get µn+1 ≥ µd+1 > µ1 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then recall that by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6 we get ΓM(t) = log(µn+1) for all t satisfying n ≤ t < n + 1 with n ∈ N such that µn+1 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In particular, as mentioned before, this holds for all n ≥ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' So let n ∈ N>0 be such that n ≥ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Take t with n ≤ t < n + 1, then one has ΣD(t) < µn+1 + 1 and log(ΣD(t)) < log(µn+1 + 1) ≤ log(eµn+1) = log(µn+1) + 1 = ΓM(t) + 1, ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS 23 showing the second part of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' On the other hand for all such t we get ΣD(t) ≥ µn+1 = exp(ΓM(t)), in fact even ΣD(t) ≥ ⌈µn+1⌉ holds, and which proves the first estimate in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ Using the counting function ΣD we set (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11) F�ΓD(x) := � |x| 0 �ΓD(s)ds, x ∈ R, with �ΓD(s) := log(ΣD(s)), s ≥ 1, �ΓD(s) := 0, 0 ≤ s < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' �ΓD is non-decreasing and right-continuous and tending to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Recall that ΣD(s) ∈ N>0 for all s ≥ 1 and this technical modification is unavoidable: In (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11) we cannot consider log(ΣD(s)) directly for all s ≥ 0 because ΣD(s) = 0 for 0 ≤ s < 1 by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then in view of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='10) we get (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='12) ∃ C, D ≥ 1 ∀ t ≥ 0 : −C + ΓM(t) ≤ �ΓD(t) ≤ ΓM(t) + D, which implies lim t→+∞ ΓM(t) �ΓD(t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Note that F�ΓD is formally not an N-function by analogous reasons as in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6: We have �ΓD(s) = 0 for (at least) all 0 ≤ s < 1, see the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Now we are able to prove the main statement of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We have the following equivalences: (i) Let M ∈ LC be given and D its dual sequence, then F�ΓD∼cϕc ωM ∼cF c M∼cϕ∗ ωM , F�ΓD ∼ ϕc ωM ∼ F c M ∼ ϕ∗ ωM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) Let G be an N-function, M G ∈ LC the associated sequence defined via (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4) and DG the corresponding dual sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Finally, let ϕG be given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then F�ΓDG ∼cϕ∗ ωMG ∼cϕc ωMG∼cF c MG∼cGc∼c(ϕG)c, F�ΓDG ∼ ϕ∗ ωMG ∼ ϕc ωMG ∼ F c MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (i) In view of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4 only F�ΓD∼cϕc ωM resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' F�ΓD ∼ ϕc ωM has to be verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' This follows analogously as in the proof of (b) in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6 (see also (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='14)) by using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='12) and the representations (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Note that these representations imply limt→+∞ F�ΓD (t) t = limt→+∞ ϕc ωM (t) t = +∞, see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) By Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5 it suffices to verify F�ΓDG ∼cϕc ωMG and F�ΓDG ∼ ϕc ωMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Both relations follow by applying the first part to M G and DG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Concerning ∼ note that here both functions are given directly by the representations (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11) resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5), so we are not involving formula (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3) and since in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='12) only additive constants appear the problem described in the proof of Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5 (see (ii) in Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2) does not occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ We remark that F�ΓD does not coincide with FD directly but nevertheless the dual sequence D can be used to get an alternative (equivalent) representation and description of the complementary N-function F c M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 24 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' SCHINDL 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Growth and regularity conditions for associated N-functions In the theory of Orlicz classes and Orlicz spaces several conditions for (abstractly given) N-functions appear frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The aim of this section is to study these known growth and regularity assump- tions in the weight sequence setting in terms of given M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We remark that all appearing conditions are naturally preserved under ∼c for (associated) N- functions, see the given citations in the forthcoming sections resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' However, by inspecting the proofs one can see that this fact also holds for a wider class of functions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' when satisfying normalization, convexity, being non-decreasing and tending to infinity (in particular for ϕωM ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Therefore recall that the technical failure of ϕωM to be formally an N-function occurs at the point 0 (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6) whereas for all crucial conditions under consideration large values t ≥ t0 > 0 are relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Of course, it makes also sense to consider the conditions in this section for arbitrary functions F : [0, +∞) → [0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The ∆2-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The most prominent property is the so-called ∆2-condition, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' [11, Chapter I, §4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 23], which reads as follows: (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1) lim sup t→+∞ F(2t) F(t) < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' This growth-condition is precisely (ω1) for F and thus also frequently appears for so-called weight functions ω in the sense of Braun-Meise-Taylor, see [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' It is straight-forward that ∆2 is preserved under relation ∼ and also under ∼c, see [11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' It is known, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' [11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2] and [1, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4], that LF is a linear space if and only if F satisfies the ∆2-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Moreover, we also get: Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let F1 and F2 be two N-functions such that either F1 or F2 has ∆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then F1⪯cF2 if and only if F2 ⪯ F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In fact, in order to conclude, we only require that either F1 or F2 is normalized and convex and that either F1 or F2 has ∆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2) we have that F1 ⪯ F2 implies F2⪯cF1 (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The converse implication holds by an iterated application of ∆2 for either F1 or F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ In the weight sequence setting in view of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11 we require (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ω1)) not for ωM directly but for ϕωM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Note that in [20, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1] we have already given a characterization of (ω1) for ωM in terms of M but ∆2 for ϕωM (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' equivalently for FM) does precisely mean (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2) lim sup t→+∞ ωM(t2) ωM(t) < +∞, which is obviously stronger than (ω1) for ωM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Concerning this requirement we formulate the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M ∈ LC be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then the following are equivalent: (i) The associated N-function FM (see Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11) satisfies the ∆2-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) ϕωM satisfies the ∆2-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iii) ωM satisfies (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS 25 (iv) ωM satisfies (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3) ∃ C > 0 ∃ H > 0 ∀ t ≥ 0 : ωM(t2) ≤ CωM(Ht) + C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (v) M satisfies (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4) ∃ k ∈ N>0 ∃ A, B ≥ 1 ∀ j ∈ N : (Mj)2k ≤ ABjMkj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3) has already appeared (crucially) in different contexts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' it is denoted by (ω7) in [9] and in [18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' by (ω8) in [15] and by Ξ in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (i) ⇔ (ii) This is clear by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11 since FM∼cϕωM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) ⇔ (iii) This is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iii) ⇒ (iv) This is clear since ωM is non-decreasing, limt→+∞ ωM(t) = +∞ and w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' H ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iv) ⇒ (iii) As mentioned at the beginning of [9, Appendix A] each non-decreasing function satis- fying (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3) has already (ω1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' By iterating this property we get ωM(t2) ≤ C1ωM(t) + C1 for some C1 ≥ 1 and all t ≥ 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iv) ⇔ (v) This has been shown in [3, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We list some examples and their consequences: (∗) According to [3, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5] we know that any M ∈ LC cannot satisfy (mg) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4) simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In particular, the Gevrey sequences Gs := (j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='s)j∈N, s > 0, are violating (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) Consider the sequences M q,n := (qjn)j∈N with q, n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4) is valid (with A = B = 1 and k satisfying k ≥ 21/(n−1)) as it is shown in [3, Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6 (1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) Combining Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2 and Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5 yields the following: Let M, L ∈ LC be given and assume that either M or L has (mg) and that either M or L has (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1) ⇐⇒ FM ⪯ FL ⇐⇒ FL⪯cFM ⇐⇒ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5) the second implication can also be reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M, L ∈ LC be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Assume that both FM and FL satisfy the ∆2-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then FM·L and FM⋆L satisfy the ∆2-condition, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' By assumption M and L both have (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4) and so it is immediate that M ·L has this property as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Concerning the convolution note that we get ωM⋆L = ωM +ωL and so ωM⋆L has (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3) because both ωM and ωL have this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ Finally we apply Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2 to M = M G and get the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let G be an N-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let ωG be the function from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1), M G ∈ LC the associated sequence defined via (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4) and FMG the associated N-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then the following are equivalent: (i) G satisfies the ∆2-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) The associated N-function FMG satisfies the ∆2-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iii) ϕωMG satisfies the ∆2-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iv) The function ϕG (see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='13)) satisfies the ∆2-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (v) ωG (equivalently ωMG) satisfies (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (vi) ωG (equivalently ωMG) satisfies (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (vii) M G satisfies (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 26 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' SCHINDL Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Everything is immediate from Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2 applied to M G: (i) ⇔ (ii) ⇔ (iii) ⇔ (iv) holds by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5 and since ∆2 is preserved under equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Concerning (v) and (vi) note that condition (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2) resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3) holds true equivalently for ωG and ωMG (by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5) and since these conditions are preserved under ∼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The ∇2-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let F be an N-function and F c its complementary function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In [11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2] it has been shown that F c satisfies the ∆2-condition if and only if F has (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5) ∃ ℓ > 1 ∃ t0 > 0 ∀ t ≥ t0 : 2ℓF(t) ≤ F(ℓt), also known under the name ∇2 for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We summarize some properties for ∇2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' For (i) and (ii) it is sufficient to require that F, G : [0, +∞) → [0, +∞) are non-decreasing, normalized and convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (i) ∇2 is preserved under relation ∼c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' this follows either from (i) in Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2 and since ∆2 is preserved under ∼c, or it can be seen directly as follows: Assume that F has ∇2 and let G be another N-function such that F∼cG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' So ∃ k > 1 ∃ t0 > 0 ∀ t ≥ t0 : G(tk−1) ≤ F(t) ≤ G(tk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' When iterating ∇2 we find (since ℓ > 1) ∃ t1 > 0 ∀ n ∈ N>0 ∀ t ≥ t1 : F(t) ≤ 1 (2ℓ)n F(ℓnt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We choose d ∈ N>0 such that ℓd ≥ k2 and n ∈ N>0 such that 2n−1 ≥ ℓd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then we estimate for all t ≥ max{t0, t1} as follows: G(t) ≤ F(kt) ≤ 1 (2ℓ)n F(ℓnkt) ≤ 1 (2ℓ)n F(ℓn+dk−1t) ≤ 1 (2ℓ)n G(ℓn+dt) ≤ 1 2ℓn+d G(ℓn+dt), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' ∇2 for G holds with ℓn+d and for all t ≥ t2 := max{t0, t1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) Similarly, we show that ∇2 is preserved under relation ∼: Assume that F has ∇2 and let G be another N-function such that F ∼ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' So ∃ k > 1 ∃ t0 > 0 ∀ t ≥ t0 : k−1G(t) ≤ F(t) ≤ kG(t), and then we take n ∈ N>0 such that k2 ≤ 2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Iterating n-times property ∇2 gives for all t sufficiently large that 1 k(2ℓ)nG(t) ≤ (2ℓ)nF(t) ≤ F(ℓnt) ≤ kG(ℓnt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Since G is an N-function we get k2G(ℓnt) ≤ G(k2ℓnt) (recall (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2)) and 2k2ℓn ≤ (2ℓ)n by the choice of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Consequently, 2k2ℓnG(t) ≤ G(k2ℓnt) is verified for all sufficiently large t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' ∇2 for G with k2ℓn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iii) Let us prove that for any non-decreasing F : [0, +∞) → [0, +∞) with limt→+∞ F(t) = +∞ we have that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5) is equivalent to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6) ∃ C ≥ 1 ∃ ℓ > 1 ∀ t ≥ 0 : 2ℓF(t) ≤ F(ℓt) + 2ℓC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5) implies (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6) with the same ℓ, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' take C := F(t0), because F is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' For the converse first we iterate (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6) and get 4ℓ2F(t) ≤ 2ℓF(ℓt)+4ℓ2C ≤ F(ℓ2t)+2ℓC +4ℓ2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then, since F(t) → +∞ as t → +∞ we can find t0 > 0 such that F(ℓ2t) ≥ 2ℓC + 4ℓ2C for all t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Thus 4ℓ2F(t) ≤ 2F(ℓ2t) for all t ≥ t0 is verified, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5) with the choice ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS 27 In the weight sequence setting we are interested in having (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5) for FM resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' for ϕωM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Since ∇2 is preserved under equivalence, via Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11 this condition transfers into (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7) ∃ ℓ > 1 ∃ s0 > 1 ∀ s ≥ s0 : 2ℓωM(s) ≤ ωM(sℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The aim is to characterize now (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7) in terms of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M ∈ LC be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then the following are equivalent: (i) The associated N-function FM satisfies the ∇2-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) ϕωM satisfies the ∇2-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iii) ωM satisfies (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iv) ωM satisfies (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8) ∃ C ≥ 1 ∃ ℓ > 1 ∀ s ≥ 0 : 2ℓωM(s) ≤ ωM(sℓ) + 2ℓC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (v) The sequence M satisfies (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='9) ∃ A ≥ 1 ∃ ℓ > 1 ∀ j ∈ N : M2j ≤ AM 2ℓ j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The proof shows that in (iv) and (v) we can take the same choice for ℓ and the correspondence between C and A is given by A = e2ℓC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Consequently, if (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8) holds for ℓ > 1 then also for all ℓ′ ≥ ℓ (even with the same choice for C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (i) ⇔ (ii) follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11 and (i) in Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) ⇔ (iii) is clear and (iii) ⇔ (iv) follows as in resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' by (iii) in Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iv) ⇒ (v) By using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5) we get for all j ∈ N: M2j = sup t≥0 t2j exp(ωM(t)) = sup t≥0 t2jℓ exp(ωM(tℓ)) ≤ e2ℓC sup t≥0 t2jℓ exp(2ℓωM(t)) = e2ℓC � sup t≥0 tj exp(ωM(t)) �2ℓ = e2ℓCM 2ℓ j , so (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='9) is verified with A := e2ℓC and the same ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (v) ⇒ (iv) We have tj Mℓ j ≤ √ A tj (M2j)1/2 for all j ∈ N and t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' This yields by definition of associated weight functions (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='10) ∀ t ≥ 0 : ωMℓ(t) ≤ ω� M2(t) + A1, with A1 := log(A) 2 , M ℓ := (M ℓ j )j∈N and the auxiliary sequence � M 2 := (M 1/2 2j )j∈N, see [21, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3], [20, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6)] and also [4, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' When taking c = 2 in [4, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7)], then (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11) ∃ D ≥ 1 ∀ t ≥ 0 : ω� M2(t) ≤ 2−1ωM(t) ≤ 2ω� M2(t) + D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4) and the first half of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11) we continue (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='10) and get ∀ t ≥ 0 : ℓωM(t1/ℓ) = ωMℓ(t) ≤ ω� M2(t) + A1 ≤ 2−1ωM(t) + A1, hence (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8) is verified with the same ℓ and C := A1 ℓ = log(A) 2ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We provide some examples of sequences such that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='9) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 28 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' SCHINDL (i) Any M ∈ LC satisfying (mg) does also have (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='9): By [18, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1] or [18, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3] applied to the matrix M = {M} (see also [13, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 1]) we know that (mg) is equivalent to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='12) ∃ B ≥ 1 ∀ j ∈ N : M2j ≤ BjM 2 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then note that BjM 2 j ≤ AM 2ℓ j ⇔ B ≤ A1/jM 2(ℓ−1)/j j holds for all j ∈ N>0 if A ≥ 1 is chosen sufficiently large because limj→+∞(Mj)1/j = +∞ and ℓ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Thus, in particular, the Gevrey sequences Gs, s > 0, have (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) However, the converse implication is not valid in general: Consider again the sequences M q,n := (qjn)j∈N with q, n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Condition (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='9) is valid because for given n > 1 we choose ℓ ≥ 2n−1 and get for all j ∈ N (and q > 1) that q(2j)n = M q,n 2j ≤ (M q,n j )2ℓ = (qjn)2ℓ ⇔ 1 ≤ q2jn(ℓ−2n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' But (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='12) is violated: This requirement means q(2j)n ≤ Bjq2jn and since n > 1 this is impossible for any choice of B as j → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M, L ∈ LC be given and assume that both FM and FL satisfy the ∇2-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then FM·L and FM⋆L have the ∇2-condition as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' By assumption both sequences satisfy (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='9) and so it is immediate that M · L has this property, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Concerning the convolution we have ωM⋆L = ωM + ωL and so ωM⋆L has (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8) because both ωM and ωL have this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (For this recall that if (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8) holds for some ℓ > 1 then for all ℓ′ ≥ ℓ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=') □ Finally, let us apply Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7 to M = M G from Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let G be an given N-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let ωG be the associated weight function from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1), M G the associated weight sequence (see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4)) and finally FMG the N-function associated with M G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then the following are equivalent: (i) G satisfies the ∇2-condition (equivalently the complementary N-function Gc satisfies the ∆2-condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) FMG satisfies the ∇2-condition (which is equivalent to the fact that the complementary N-function F c MG satisfies the ∆2-condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iii) ϕωMG satisfies the ∇2-condition (equivalently the complementary function ϕc ωMG satisfies the ∆2-condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iv) The function ϕG (see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='13)) satisfies the ∇2-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (v) ωG (equivalently ωMG) satisfies (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (vi) ωG (equivalently ωMG) satisfies (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (vii) M G satisfies (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We apply Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7 to M G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Again (i) ⇔ (ii) ⇔ (iii) ⇔ (iv) follows by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5 and the fact that both the ∆2 and the ∇2-condition are preserved under equivalence, recall (i) in Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6 for the latter one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Concerning (v) and (vi) note that by (ii) in Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6 both (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8) are preserved under relation ∼ which holds between ωG and ωMG (recall (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS 29 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The ∆2-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' According to [11, Chapter I, §6, 5] we say that an N-function F satisfies the ∆2-condition if ∃ k > 1 ∃ t0 > 0 ∀ t ≥ t0 : F(t)2 ≤ F(kt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' As stated on [11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 41], the ∆2-condition is preserved under relation ∼c and ∆2 holds if and only if F α∼cF for some/any α > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In fact, this equivalence holds for any non-decreasing F : [0, +∞) → [0, +∞) with limt→+∞ F(t) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' By [11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8] we know that F has ∆2 if and only if the complementary function F c has ∃ k > 1 ∃ t0 > 0 ∀ t ≥ t0 : F c(t2) ≤ ktF c(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In view of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11 the associated N-function FM satisfies the ∆2-condition if and only if ∃ k > 1 ∃ t0 > 0 ∀ t ≥ t0 : (ωM(et))2 = ϕωM (t)2 ≤ ϕωM (kt) = ωM(ekt), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='13) ∃ k > 1 ∃ s0 > 1 ∀ s ≥ s0 : ωM(s)2 ≤ ωM(sk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Note that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='13) is not well-related to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5) and in general it seems to be difficult to obtain a characterization for ∆2 in terms of M by using this formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' However, we give a sufficient condition to ensure ∆2 in the weight sequence setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' First we prove the following technical result: Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M ∈ LC be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then the following are equivalent: (i) The counting function ΣM satisfies (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='14) ∃ K > 0 ∃ t0 > 0 ∀ t ≥ t0 : ΣM(et)2 ≤ ΣM(etK), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' [11, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='9)] for p = ΣM ◦ exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) The sequence of quotients µ satisfies (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='15) ∃ A > 0 ∃ j0 ∈ N ∀ j ≥ j0 : µj2 ≤ µA j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The proof shows the correspondence A = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (i) ⇒ (ii) Write s := et and so ΣM(s)2 ≤ ΣM(sK) is satisfied for all s ≥ s0 := et0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let j0 ∈ N>0 be minimal satisfying µj0 ≥ s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let j ≥ j0 be such that µj < µj+1 and take s with µj ≤ s < µj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then j2 = ΣM(s)2 ≤ ΣM(sK) follows which implies µj2 ≤ sK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In particular, when taking s := µj we have shown (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='15) with A := K and all j ≥ j0 such that µj < µj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' If j ≥ j0 with µj = · · · = µj+ℓ < µj+ℓ+1 for some ℓ ∈ N>0, then following the previous step we get µ(j+ℓ)2 ≤ µK i for all j ≤ i ≤ j + ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Since (j + ℓ)2 ≥ i2 for all such indices i and since by log-convexity j �→ µj is non-decreasing we are done for all j ≥ j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) ⇒ (i) Let s ≥ µj0 and so µj ≤ s < µj+1 for some j ≥ j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then ΣM(s) = j and sA ≥ µA j ≥ µj2 which implies ΣM(sA) ≥ j2 = ΣM(s)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Thus (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='14) is shown with K := A and t0 = log(µj0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ Using this result we get the following: Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M ∈ LC be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Assume that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='15) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then the associated N- function FM (or equivalently ϕωM ) satisfies the ∆2-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11 we have that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='14) is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then in view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='17) the function fM appearing in the representation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1) of FM enjoys (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='14) (with the same K and for all t ≥ t1, t1 possibly strictly larger than t0 appearing in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='14)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Thus we can apply [11, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4] 30 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' SCHINDL in order to conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (There the estimate in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='14) is assumed to be strict;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' however the proof only requires ≤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=') □ Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let G be an N-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' If the associated weight sequence M G (see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4)) satisfies (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='15), then G and the associated N-function FMG and ϕG (see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='13)) enjoy the ∆2-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Recall that in order to verify (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='15) for M G for abstractly given N-functions G the formula 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='10 can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' First, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='12 applied to M G yields ∆2 for FMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5 we have that FMG, G and ϕG are equivalent and since ∆2 is preserved under equivalence we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We gather some observations: (i) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='15) means that the sequence of quotients has to increase ”relatively slowly” and w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' we can assume A ∈ N≥2 in this condition (since µj ≥ 1 for all j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='15) is preserved under relation ∼=: Let M, L ∈ LC such that M ∼= L and assume that M has (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then ∃ A > 0 ∃ B ≥ 1 ∃ j0 ∈ N ∀ j ≥ j0 : 1 B λj2 ≤ µj2 ≤ µA j ≤ BλA j follows and since limj→+∞ λj = +∞ we get B2λA j ≤ λA′ j for any A′ > A and for all j ≥ jA′,B sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Thus L satisfies (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='15) when choosing A′ > A and restricting to j ≥ max{j0, jA′,B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iii) In [11, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='10), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 43] another sufficiency criterion is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Suppose that an N-function F has ∃ α > 0 ∃ t0 > 0 : t �→ log(F(t)) tα is not decreasing on [t0, +∞), then F satisfies the ∆2-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' One verifies that this condition yields F 2∼cF and hence this implication holds for any non-decreasing F : [0, +∞) → [0, +∞) with limt→+∞ F(t) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In the weight sequence setting this expression amounts to the study of log(ϕωM (t)) tα = log(ωM(et)) tα = log(ωM(s)) log(s)α for all s ≥ s0 = et0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' If there exists α > 0 such that s �→ log(ωM(s)) log(s)α is non-decreasing for all large s, then ϕωM satisfies the ∆2-condition and so FM as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We comment on some examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) All Gevrey-sequences Gs, s > 0, satisfy (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='15): This condition amounts to (j2)s ≤ (js)A and so the choices A := 2 and j0 := 0 are sufficient (for any s > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) If M, L ∈ LC both have (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='15), then also the product sequence M · L: The corresponding sequence of quotients is given by the product µ · λ and so (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='15) follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The same implication is not clear for the convolution product M ⋆ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) The sequence M q,2 does not satisfy this requirement because in this case the corresponding sequence of quotients is given by (q2j−1)j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='15) transfers into q2j2−1 ≤ q(2j−1)A but which is impossible for any choice A ≥ 1 if j → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' And in this case also (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='13) is violated: We have, see the proof of Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='20 for more details and citations, that ωMq,2 ∼ ω2 for all q > 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' ωMq,2(t) = O(ω2(t)) and ω2(t) = O(ωMq,2(t)) as t → +∞ for each q > 1 with ω2(t) := max{0, log(t)2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Fix q > 1 and so (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='13) gives for some C ≥ 1 and all s ≥ 1 −1 + C−1 log(s)4 ≤ ωMq,2(s)2 ≤ ωMq,2(sk) ≤ C log(sk)2 + C = Ck2 log(s)2 + C, ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS 31 yielding a contradiction as s → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Summarizing, by Examples 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='15 we get the following consequences: (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='16) ∆2 ∧ ∇2, ∆2, ∇2 ⇏ ∆2, ∆2 ⇏ ∆2, ∇2 ⇏ ∆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let us study how condition (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='15) is related to moderate growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M ∈ LC be given such that (mg) holds and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='17) lim inf n→+∞(µ2n)1/(n+2) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then M satisfies (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='15) and hence the associated N-function FM (or equivalently ϕωM ) satisfies the ∆2-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' First, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' [16, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2] and the citations there, (mg) is equivalent to supj∈N µ2j µj < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Consequently, we find some A > 1 such that µ2kj ≤ Akµj for each k, j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Take now j ∈ N>0 (the case j = 0 is trivial), then 2n ≤ j < 2n+1 for some n ∈ N and 22n ≤ j2 < 22n+2, so µj2 ≤ µ22n+2 = µ2n+22n ≤ An+2µ2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' By (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='17) we find n0 ∈ N and δ > 1 such that (µ2n)1/(n+2) ≥ δ for all n ≥ n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Set B := log(A) log(δ) +1 > 1, so δ = A1/(B−1) and hence An+2 ≤ µB−1 2n for all n ≥ n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' This implies µj2 ≤ An+2µ2n ≤ µB 2n ≤ µB j and so (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='15) is verified (for j0 := 2n0 and choosing B as before).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ We finish by commenting on requirement (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='17): (∗) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='17) is a mild extra growth assumption: It follows when lim infj→∞ µj j > 0 because then µj ≥ jǫ for some ǫ > 0 and all j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Thus (µ2n)1/(n+2) ≥ 2n/(n+2)ǫ1/(n+2) for all n ∈ N and so (µ2n)1/(n+2) > 1 for all n sufficiently large (depending on ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) On the other hand note that limn→+∞(2ns)1/(n+2) = 2s > 1 for any s > 0 and hence each Gs satisfies (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' However, limj→+∞ js j = 0 holds for all 0 < s < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The ∆3-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' An N-function F satisfies the ∆3-condition, see [11, Chapter I, §6, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1)], if ∃ k > 1 ∃ t0 > 0 ∀ t ≥ t0 : tF(t) ≤ F(kt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Since F(t) ≥ t for all large t (recall the second part in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3)) we immediately have that ∆2 implies ∆3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' however the converse is not true in general, see [11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' It is also known that ∆3 for F implies ∆2 for F c, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' F has ∇2, see [11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' ∆3 is preserved under equivalence and since tF(t) ≥ F(t) for all t ≥ 1 we get that (∗) F has ∆3 if and only if (∗) F and t �→ tF(t) are equivalent, see [11, Chapter I, §6, 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Moreover, we have the following reformulation for ∆3: Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let F be an N-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then the following are equivalent: (i) F satisfies the ∆3-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) We have �F∼cF with (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='18) �F(t) := � |t| 0 F(s)ds, t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Consequently, if any of these equivalent conditions holds true, then �F has ∆3, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 32 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' SCHINDL Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (i) ⇒ (ii) This is contained in the proof of [11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' First, for any N-function F we get for all t ≥ 1 that �F(2t) = � 2t 0 F(s)ds ≥ � 2t t F(s)ds ≥ F(t)t ≥ F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In fact this holds for any non-decreasing and non-negative F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' On the other hand, by using ∆3 for some k > 1 and all t sufficiently large one has �F(t) = � t 0 F(s)ds ≤ F(t)t ≤ F(kt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) ⇒ (i) By the equivalence we get �F(t) ≤ F(kt) for some k > 1 and all t sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Thus for all such large t we estimate by F(k2t) ≥ �F(2t) = � 2t 0 F(s)ds = � t 0 F(s)ds + � 2t t F(s)ds ≥ �F(t) + F(t)t ≥ F(t)t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' ∆3 with k′ := 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ We apply this characterization to the weight sequence setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M ∈ LC be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then the following are equivalent: (i) The associated N-function FM (equivalently ϕωM ) satisfies the ∆3-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) �FM∼cFM holds true (with �FM given by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='18)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iii) �ϕωM ∼cϕωM holds true with �ϕωM (t) := � |t| 0 ϕωM (s)ds, t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iv) We have that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='19) ∃ k > 1 ∃ s0 > 1 ∀ s ≥ s0 : ωM(s) ≤ �ωM(s2) ≤ ωM(sk), with �ωM(t) := �ϕωM (log(t)), t ≥ 1, �ωM(t) := 0, 0 ≤ t < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The function �ωM admits the representation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='20) �ωM(s) = � |s| 0 ωM(u) u du = � |s| 1 ωM(u) u du, s ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (i) ⇔ (ii) follows by Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='17 applied to FM and the fact that ∆3 is preserved under equivalence, see Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) ⇒ (iii) The estimate �ϕωM (2t) ≥ ϕωM (t) for t ≥ 1 holds as in (i) ⇒ (ii) in Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='17 and for the converse we estimate as follows for t sufficiently large: �ϕωM (t) = � t 0 ϕωM (s)ds ≤ � t 0 (FM(s) + C)ds = �FM(t) + Ct ≤ FM(kt) + Ct ≤ ϕωM (kt) + Ct + D ≤ 2ϕωM (kt) ≤ ϕωM (2kt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The first estimate follows for some C ≥ 1 (and all s ≥ 0) by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='16), the second one since �FM∼cFM by assumption, the third one again by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='16), the fourth since limt→+∞ ϕωM (t) t = +∞, and finally the last one by convexity and normalization for ϕωM (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS 33 (iii) ⇒ (ii) �FM(2t) ≥ FM(t) for all t ≥ 1 is shown in (i) ⇒ (ii) in Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Conversely, by assumption �ϕωM (t) ≤ ϕωM (kt) for some k > 1 and all t(≥ 1) large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Thus for all t sufficiently large: �FM(t) = � t 0 FM(s)ds ≤ � t 0 (ϕωM (s) + D)ds = �ϕωM (t) + Dt ≤ ϕωM (kt) + Dt ≤ FM(kt) + Dt + C ≤ 2FM(kt) ≤ FM(2kt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The first estimate follows by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='16) (for all s ≥ 0), the second one by assumption, the third one again by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='16), for the fourth estimate we have used the second part in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3) and finally the last one holds by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iii) ⇔ (iv) First, for all t ≥ 0 we have �ωM(et) = �ϕωM (t) = � t 0 ωM(es)ds = � et 1 ωM(u) u du = � et 0 ωM(u) u du, because ωM(t) = 0 for 0 ≤ t ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Thus �ωM(t) = � t 0 ωM(u) u du for all t ≥ 1 and in fact even for all t ≥ 0 (since �ωM(t) := 0 for 0 ≤ t ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Thus (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='20) is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Moreover, �ϕωM ∼cϕωM holds if and only if (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='21) ∃ k > 1 ∃ t0 > 0 ∀ t ≥ t0 : ϕωM (t) ≤ �ϕωM (2t) ≤ ϕωM (kt), since the first estimate holds for all t ≥ 1 as mentioned in (ii) ⇒ (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='21) is obviously equivalent to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ Using this characterization we give two applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M, L ∈ LC be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' If both FM and FL have the ∆3-condition, then FM⋆L, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Recall that M ⋆ L ∈ LC and ωM⋆L = ωM + ωL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='20) implies �ωM⋆L = �ωM + �ωL and by assumption we have (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='19) for both ωM and ωL and so for ωM⋆L as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='18 yields that FM⋆L satisfies the ∆3-condition, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' There exist N-functions satisfying ∆2 and ∇2 but not ∆3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let us consider the sequence(s) M q,n with q, n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' As seen in Examples 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='8 each sequence yields an associated N-function FMq,n satisfying both ∆2 and ∇2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We prove now that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='19) is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' For this first recall results from [16, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5] and [18, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='10]: Let n > 1 be arbitrary but from now on fixed, then each M q,n is an element of the weight matrix (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' the one-parameter family of weight sequences) associated with the weight ωs(t) := max{0, log(t)s}, s > 1, such that 1 s + 1 n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Therefore, ωMq,n ∼ ωs for all q > 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' ωMq,n(t) = O(ωs(t)) and ωs(t) = O(ωMq,n(t)) as t → +∞ for each q > 1, see [18, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='3] and [15, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' For all x ≥ 1 we have � x 0 ωs(t) t dt = � x 1 log(t)s t dt = �log(t)s+1 s + 1 �t=x t=1 = log(x)s+1 s + 1 , and so by the above (recall also (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='20)) ∀ q > 1 ∃ D ≥ 1 ∀ x ≥ 1 : D−1 log(x)s+1 s + 1 − log(x) ≤ �ωMq,n(x) ≤ Dlog(x)s+1 s + 1 + D log(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 34 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' SCHINDL Fix now q > 1 and then, when (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='19) is valid, we obtain D−1 log(x2)s+1 s + 1 − log(x2) ≤ �ωMq,n(x2) ≤ ωMq,n(xk) ≤ D log(xk)s + D =⇒ 2s+1 log(x)s+1 ≤ D2(s + 1)ks log(x)s + D2(s + 1) + 2(s + 1)D log(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' But this is impossible as x → +∞ for any choices of D and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ By applying Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='18 to the associated sequence M G and recalling that ∆3 is preserved under equivalence we formulate the last statement in this section: Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let G be an N-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then the following are equivalent: (i) G satisfies the ∆3-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (ii) The associated N-function FMG satisfies the ∆3-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iii) ϕG (see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='13)) satisfies the ∆3-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (iv) The other assertions listed in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='18 are valid for the associated sequence M G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' The ∆′-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' According to [11, Chapter I, §5] we say that an N-function F satisfies the ∆′-condition if ∃ k > 0 ∃ u0 > 0 ∀ t, s ≥ u0 : F(ts) ≤ kF(t)F(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In the weight sequence setting in view of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='11 and since ∆′ is preserved under equivalence, see [11, Chapter I, §5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 30], this condition means that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='22) ∃ k > 0 ∃ u0 > 0 ∀ t, s ≥ u0 : ωM(tlog(s)) ≤ kωM(s)ωM(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' By [11, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1] we know that ∆′ implies ∆2 and on [11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 30-31] it is shown that in general this implication is strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Moreover, by [11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='6] it follows that if F satisfies ∆2, then F c has ∆′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' A direct check of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='22) seems to be quite technical resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' hardly possible since this estimate is not well-related w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In [11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1] a sufficiency criterion for ∆′ is shown: Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let F(t) = � |t| 0 f(s)ds be a given N-function (recall the representation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then F satisfies the ∆′-condition provided that f has the following growth property which we abbreviate by (∆′ f) from now on: There exists some t0 > 1 such that for every fixed t ≥ t0 the function hf given by hf(s) := f(st) f(s) is not increasing on [t0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In the weight sequence setting this result takes the following form: Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M ∈ LC be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' If ΣM ◦ exp satisfies (∆′ ΣM ◦exp) then the associated N- function FM (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' equivalently ϕωM ) satisfies the ∆′-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='17), we get (∆′ ΣM ◦exp) if and only if (∆′ fM ) when enlarging t0 sufficiently if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='22 applied to f = fM and F = FM yields the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ However, we show now that in general (∆′ ΣM ◦exp) fails in the weight sequence setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let M ∈ LC be given such that 1 ≤ µ1 < µ2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' the sequence of quotients is strictly increasing (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then (∆′ ΣM ◦exp) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' equivalently (∆′ fM )) is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' ON ORLICZ CLASSES DEFINED IN TERMS OF ASSOCIATED WEIGHT FUNCTIONS 35 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' First, with u := es we get ΣM (ets) ΣM(es) = ΣM(ut) ΣM(u) and hence (∆′ ΣM ◦exp) precisely means: ∃ t0 > 1 ∀ t ≥ t0 : u �→ ΣM(ut) ΣM(u) is not increasing on [et0, +∞) =: [u0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let now t ≥ t0 > 1 be arbitrary but fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then µj0 ≤ u0 < µj0+1 and µk ≤ ut 0 < µk+1 for some (large) j0, k ∈ N>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Note that k is depending on t and ΣM(ut 0) ΣM(u0) = k j0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Since ut 0 ≥ u0 we clearly have k ≥ j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We show that assuming (∆′ ΣM ◦exp) yields a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Let u ∈ [u0, +∞) increase and we split the argument in several steps: (∗) If u ≥ u0 is given with µj0 < u < µj0+1 and µk < ut < µk+1, then for all u′ ∈ [ǫ − u, u + ǫ] with ǫ > 0 sufficiently small the quotient appearing in (∆′ ΣM ◦exp) remains constant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' in this case the crucial expression is locally constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) Even k > j0 is valid: If k = j0, then in order to have (∆′ ΣM◦exp) we need µj0 ≤ u < ut < µj0+1 for all u ≥ u0 with µj0 < u < µj0+1, a contradiction as u → µj0+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) If (∆′ ΣM ◦exp) holds true, then for all u with µj0 ≤ u0 ≤ u < µj0+1 it follows that ut < µk+1: Otherwise, if ut ≥ µk+1, then ΣM(ut) ΣM (u) ≥ k+1 j0 > k j0 = ΣM (ut 0) ΣM (u0), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) On the other hand, for all u ≥ u0 satisfying µk ≤ ut < µk+1 it is allowed that u ≥ µj0+d for some d ∈ N>0: In this case, if µj0+d ≤ u < µj0+d+1 and still µk ≤ ut < µk+1, then ΣM(ut) ΣM (u) = k j0+d < k j0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) Take u = (µk+1)1/t and so u > u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We get that µj0+d ≤ u < µj0+d+1 for some d ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Thus ΣM(ut) = k + 1 holds (since µ is strictly increasing!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=') and ΣM(u) = j0 + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' We distinguish: If µj0+d < u < µj0+d+1, then we can find ǫ > 0 sufficiently small to ensure u − ǫ > µj0+d and µk ≤ (u − ǫ)t < µk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In this case ΣM (ut) ΣM(u) = k+1 j0+d > k j0+d = ΣM((u−ǫ)t) ΣM (u−ǫ) , a contradiction to (∆′ ΣM◦exp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' In particular this case happens if d = 0 because then µj0 ≤ u0 < u by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) If now µj0+d = u < µj0+d+1 for some d ≥ 1 and ut = µk+1, then take ǫ > 0 sufficiently small to ensure u − ǫ > µj0+d−1 and µk ≤ (u − ǫ)t < µk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Both estimates are possible since the sequence µ is assumed to be strictly increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Thus ΣM(ut) ΣM(u) = k+1 j0+d ≤ k j0+d−1 = ΣM ((u−ǫ)t) ΣM (u−ǫ) and which verifies (∆′ ΣM◦exp) in this case because this estimate is equivalent to j0 + d − 1 ≤ k and this is clear since µj0+d = u < ut = µk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) Summarizing all the information, a necessary condition to ensure (∆′ ΣM ◦exp) is that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='23) ∃ j0 ∈ N>0 ∃ t0 > 1 ∀ t ≥ t0 ∃ k ∈ N>0, k > j0, ∃ d ∈ N>0 : µk+1 = (µj0+d)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' (∗) The equality precisely means t = log(µk+1) log(µj0+d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' However, the expression on the right-hand side can only take countable many values whereas t is required to belong to an uncountable set and therefore (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='23) is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ We finish with the following consequence showing that in general [11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1] does not provide a characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' There exist N-functions F such that F satisfies the ∆′-condition but (∆′ f) fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' 36 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' SCHINDL Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Consider the function(s) ωs, s > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Then ϕωs(t) = ts for all t ≥ 0 and so the ∆′-condition is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Since ωMq,n ∼ ωs for all q > 1 and n > 1 such that 1 s + 1 n = 1 we get that ϕωMq,n ∼ ϕωs as well (see the proof of (ii) ⇔ (iii) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' It is immediate that the ∆′-condition is also preserved under ∼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Hence ϕωMq,n and finally FMq,n satisfy the ∆′-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' However, for any q > 1 the corresponding sequence of quotients is clearly strictly increasing (recall that µq,n j = qjn−(j−1)n, j ≥ 1) and so by Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content='24 property (∆′ fMq,n ) fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' □ References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Alexopoulos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' A brief introdcution to N-functions and Orlicz function spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=' Kent State Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FJT4oBgHgl3EQfoCxt/content/2301.11594v1.pdf'} +page_content=', 2004, available online at https://dokumen.' metadata={'source': 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