diff --git "a/4tE0T4oBgHgl3EQfvQHn/content/tmp_files/load_file.txt" "b/4tE0T4oBgHgl3EQfvQHn/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/4tE0T4oBgHgl3EQfvQHn/content/tmp_files/load_file.txt" @@ -0,0 +1,553 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf,len=552 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='02617v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='FA] 6 Jan 2023 LINEAR TOPOLOGICAL INVARIANTS FOR KERNELS OF DIFFERENTIAL OPERATORS BY SHIFTED FUNDAMENTAL SOLUTIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' DEBROUWERE1 AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' KALMES2 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We characterize the condition (Ω) for smooth kernels of partial differen- tial operators in terms of the existence of shifted fundamental solutions satisfying certain properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' The conditions (PΩ) and (PΩ) for distributional kernels are characterized in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' By lifting theorems for Fr´echet spaces and (PLS)- spaces, this provides characterizations of the problem of parameter dependence for smooth and distributional solutions of differential equations by shifted fundamen- tal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' As an application, we give a new proof of the fact that the space {f ∈ E (X) | P(D)f = 0} satisfies (Ω) for any differential operator P(D) and any open convex set X ⊆ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Keywords: Partial differential operators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Fundamental solutions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Linear topological invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' MSC 2020: 46A63, 35E20, 46M18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Introduction In their seminal work [16] Meise, Taylor and Vogt characterized the constant co- efficient linear partial differential operators P(D) = P(−i ∂ ∂x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' , −i ∂ ∂xd) that have a continuous linear right inverse on E (X) and/or D′(X) (X ⊆ Rd open) in terms of the existence of certain shifted fundamental solutions of P(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Later on, Frerick and Wengenroth [8,27] gave a similar characterization of the surjectivity of P(D) on E (X), D′(X), and D′(X)/E (X) as well as of the existence of right inverses of P(D) on the latter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Roughly speaking, these results assert that P(D) satisfies some condition (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' being surjective on E (X)) if and only if for each compact subset K of X and ξ ∈ X far enough away from K there is a shifted fundamental solution E for δξ such that E satisfies a certain property on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Of course, this property depends on the condition one wants to characterize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Results of the same type have also been shown for spaces of non-quasianalytic ultradifferentiable functions and ultradistributions [13,14] and for spaces of real analytic functions [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' The aim of this paper is to complement 1Department of Mathematics and Data Science, Vrije Universiteit Brussel, Plein- laan 2, 1050 Brussels, Belgium 2Faculty of Mathematics, Chemnitz University of Technology, 09107 Chemnitz, Germany E-mail addresses: andreas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='debrouwere@vub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='be, thomas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='kalmes@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='tu-chemnitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='de.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' 1 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' DEBROUWERE AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' KALMES the above results by characterizing several linear topological invariants for smooth and distributional kernels of P(D) by means of shifted fundamental solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' The study of linear topological invariants for kernels of P(D) goes back to the work of Petzsche [19] and Vogt [23] and was reinitiated by Bonet and Doma´nski [1, 2, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' It is motivated by the question of surjectivity of P(D) on vector-valued function and distribution spaces, as we now proceed to explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We assume that the reader is familiar with the condition (Ω) for Fr´echet spaces [18] and the conditions (PΩ) and (PΩ) for (PLS)-spaces [1, 5] (see also the preliminary Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Set EP(X) = {f ∈ E (X) | P(D)f = 0} and D′ P(X) = {f ∈ D′(X) | P(D)f = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Suppose that P(D) is surjective on E (X), respectively, D′(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Given a locally convex space E, it is natural to ask whether P(D) : E (X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' E) → E (X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' E), respectively, P(D) : D′(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' E) → D′(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' E) is still surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' If E is a space of functions or distributions, this question is a reformulation of the well-studied problem of parameter dependence for solutions of partial differential equations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' see [1, 2, 5] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' The splitting theory for Fr´echet spaces [24] implies that the mapping P(D) : E (X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' E) → E (X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' E) for E = D′(Y ) (Y ⊆ Rn open) or S ′(Rn) is surjective if and only if EP(X) satisfies (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Similarly, as an application of their lifting results for (PLS)-spaces, Bonet and Doma´nski showed that the mapping P(D) : D′(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' E) → D′(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' E) for E = D′(Y ) or S ′(Rn) is surjective if and only if DP(X) satisfies (PΩ) [1], while it is surjective for E = A (Y ) if and only if D′ P(X) satisfies (PΩ) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Petzsche [19] showed that EP(X) satisfies (Ω) for any convex open set X1, while Vogt proved that this is the case for an arbitrary open set X if P(D) is elliptic [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Similarly, D′ P(X) satisfies (PΩ) for any convex open set X [1] and for an arbitrary open set X if P(D) is elliptic [1, 9, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' On the negative side, the second author [12] constructed a differential operator P(D) and an open set X ⊆ Rd such that P(D) is surjective on D′(X) (and thus also on E (X)) but EP(X) and D′ P(X) do not satisfy (Ω), respectively, (PΩ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Furthermore, D′ P(X) does not satisfy (PΩ) for any convex open set X if P(D) is hypoelliptic and for an arbitrary open set X if P(D) is elliptic [5,25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We refer to [3,5] and the references therein for further results concerning (Ω) for EP(X) and (PΩ) and (PΩ) for DP(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Apart from this classical application to the problem of surjectivity of P(D) on spaces of vector-valued smooth functions and distributions, in our recent article [4], the linear topological invariant (Ω) for EP(X) played an important role to establish quantitative approximation results of Runge type for several classes of partial differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' See [7,20,21] for other works on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' In the present note, we characterize the condition (Ω) for EP(X) and the conditions (PΩ) and (PΩ) for D′ P(X) in terms of the existence of certain shifted fundamental solutions for P(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' By the above mentioned results from [1, 5], the latter provides characterizations of the problem of distributional and real analytic parameter depen- dence for distributional solutions of the equation P(D)f = g by shifted fundamental solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' This answers a question of Doma´nski [6, Problem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='5] for distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' 1Petzsche actually showed this result under the additional hypothesis that P(D) is hypoelliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' However, as observed in [3], a careful inspection of his proof shows that this hypothesis can be omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' LINEAR TOPOLOGICAL INVARIANTS BY SHIFTED FUNDAMENTAL SOLUTIONS 3 We now state our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Set N = {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Let Y ⊆ Rd be relatively compact and open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' For N ∈ N we define ∥f∥Y ,N = max x∈Y ,|α|≤N |f (α)(x)|, f ∈ CN(Y ), and ∥f∥∗ Y ,N = sup{|⟨f, ϕ⟩| | ϕ ∈ DY , ∥ϕ∥Y ,N ≤ 1}, f ∈ (DY )′, where DY denotes the Fr´echet space of smooth functions with support in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Let P ∈ C[ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' , ξd], let X ⊆ Rd be open, and let (Xn)n∈N be an exhaustion by relatively compact open subsets of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' (a) P(D) : E (X) → E (X) is surjective and EP(X) satisfies (Ω) if and only if ∀ n ∈ N ∃ m ≥ n, N ∈ N ∀ k ≥ m, ξ ∈ X\\Xm ∃ K ∈ N, s, C > 0 ∀ ε ∈ (0, 1) ∃ Eξ,ε ∈ D′(Rd) with P(D)Eξ,ε = δξ in Xk such that ∥Eξ,ε∥∗ Xn,N ≤ ε and ∥Eξ,ε∥∗ Xk,K ≤ C εs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1) (b) P(D) : D′(X) → D′(X) is surjective and D′ P(X) satisfies (PΩ) if and only if ∀ n ∈ N ∃ m ≥ n ∀ k ≥ m, N ∈ N, ξ ∈ X\\Xm ∃ K ∈ N, s, C > 0 ∀ ε ∈ (0, 1) ∃ Eξ,ε ∈ D′(Rd) ∩ CN(Xn) with P(D)Eξ,ε = δξ in Xk such that ∥Eξ,ε∥Xn,N ≤ ε and ∥Eξ,ε∥∗ Xk,K ≤ C εs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='2) (c) P(D) : D′(X) → D′(X) is surjective and D′ P(X) satisfies (PΩ) if and only if (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='2) with “∃ K ∈ N, s, C > 0′′ replaced by “∀s > 0 ∃ K ∈ N, C > 0′′ holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1 will be given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Interestingly, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1 is somewhat of a different nature than the above mentioned results from [8,16,27] in the sense that the characterizing properties on the shifted fundamental solutions Eξ,ε are not only about the behavior of Eξ,ε on the Xn but also on the larger set Xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' In this regard, we mention that P(D) is surjective on E (X), respectively, D′(X) if and only if (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1), respectively, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='2) without the assumption ∥Eξ,ε∥∗ Xk,K ≤ C εs holds [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' It would be interesting to evaluate the conditions in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1 in specific cases in order to obtain concrete necessary and sufficient conditions on X and P for EP(X) to satisfy (Ω) and for D′ P(X) to satisfy (PΩ) and (PΩ) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' [14, 16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We plan to study this in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' As a first result in this direction, we show in Section 4 that EP(X) satisfies (Ω) for any differential operator P(D) and any open convex set X by combining Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1(a) with a powerful method to construct fundamental solutions due to H¨ormander [10, Proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' As mentioned above, this result is originally due to Petzsche [19], who proved it with the aid of the fundamental principle of Ehrenpreis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' A completely different proof was recently given by the authors in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Finally, we would like to point out that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1 implies that surjectivity of P(D) on E (X) and (Ω) for EP(X) as well as surjectivity of P(D) on D′(X) and (PΩ), respectively, (PΩ), for D′ P(X) are preserved under taking finite intersections of open 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' DEBROUWERE AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' KALMES sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' For (PΩ) this also follows from [1, Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='3] and the fact that surjectivity of P(D) is preserved under taking finite intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' However, for (Ω) and (PΩ) we do not see how this may be shown without Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Linear topological invariants In this preliminary section we introduce the linear topological invariants (Ω) for Fr´echet spaces and (PΩ) and (PΩ) for (PLS)-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We refer to [1, 5, 18] for more information about these conditions and examples of spaces satisfying them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Throughout, we use standard notation from functional analysis [18] and distribution theory [10,22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' In particular, given a locally convex space E, we denote by U0(E) the filter basis of absolutely convex neighborhoods of 0 in E and by B(E) the family of all absolutely convex bounded sets in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Projective spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' A projective spectrum (of locally convex spaces) E = (En, ̺n n+1)n∈N consists of locally convex spaces En and continuous linear maps ̺n n+1 : En+1 → En, called the spectral maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We define ̺n n = idEn and ̺n m = ̺n n+1 ◦ · · · ◦ ̺m−1 m : Em → En for n, m ∈ N with m > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' The projective limit of E is defined as Proj E = � (xn)n∈N ∈ � n∈N En | xn = ̺n n+1(xn+1), ∀n ∈ N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' For n ∈ N we write ̺n : Proj E → En, (xj)j∈N �→ xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We always endow Proj E with its natural projective limit topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' For a projective spectrum E = (En, ̺n n+1)n∈N of Fr´echet spaces, the projective limit Proj E is again a Fr´echet space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We will implicitly make use of the derived projective limit Proj1 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We refer to [26, Sections 2 and 3] for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' In particular, see [26, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='4] for an explicit definition of Proj1 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' The condition (Ω) for Fr´echet spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' A Fr´echet space E is said to satisfy the condition (Ω) [18] if ∀U ∈ U0(E) ∃V ∈ U0(E) ∀W ∈ U0(E) ∃s, C > 0 ∀ε ∈ (0, 1) : V ⊆ εU + C εsW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' The following result will play a key role in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' [3, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='4] Let E = (En, ̺n n+1)n∈N be a projective spectrum of Fr´echet spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Then, Proj1 E = 0 and Proj E satisfies (Ω) if and only if ∀n ∈ N, U ∈ U0(En) ∃m ≥ n, V ∈ U0(Em) ∀k ≥ m, W ∈ U0(Ek) ∃s, C > 0 ∀ε ∈ (0, 1) : ̺n m(V ) ⊆ εU + C εs̺n k(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1) LINEAR TOPOLOGICAL INVARIANTS BY SHIFTED FUNDAMENTAL SOLUTIONS 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' The conditions (PΩ) and (PΩ) for (PLS)-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' A locally convex space E is called a (PLS)-space if it can be written as the projective limit of a spectrum of (DFS)-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Let E = (En, ̺n n+1)n∈N be a spectrum of (DFS)-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We call E strongly reduced if ∀n ∈ N ∃m ≥ n : ̺n m(Em) ⊆ ̺n(Proj E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' The spectrum E is said to satisfy (PΩ) if ∀n ∈ N ∃m ≥ n ∀k ≥ m ∃B ∈ B(En) ∀M ∈ B(Em) ∃K ∈ B(Ek), s, C > 0 ∀ε ∈ (0, 1) : ̺n m(M) ⊆ εB + C εs̺n k(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='2) The spectrum E is said to satisfy (PΩ) if (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='2) with “∃K ∈ B(Ek), s, C > 0” replaced by “∀s > 0 ∃K ∈ B(Ek), C > 0” holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' A (PLS)-space E is said to satisfy (PΩ), respectively, (PΩ) if E = Proj E for some strongly reduced spectrum E of (DFS)-spaces that satisfies (PΩ), respectively, (PΩ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' This notion is well-defined as [26, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='8] yields that all strongly reduced projective spectra E of (DFS)-spaces with E = Proj E are equivalent (in the sense of [26, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' The bipolar theorem and [1, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='5] imply that the above definitions of (PΩ) and (PΩ) are equivalent to the original ones from [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1 This section is devoted to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We fix P ∈ C[ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' , ξd]\\{0}, an open set X ⊆ Rd, and an exhaustion by relatively compact open subsets (Xn)n∈N of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' For n, N ∈ N we write ∥ · ∥n,N = ∥ · ∥Xn,N and ∥ · ∥∗ n,N = ∥ · ∥∗ Xn,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' For ξ ∈ Rd and r > 0 we denote by B(ξ, r) the open ball in Rd with center ξ and radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Moreover, for p ∈ {1, ∞} and N ∈ N we set ∥ϕ∥Lp,N = max |α|≤N ∥ϕ(α)∥Lp, ϕ ∈ D(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We fix χ ∈ D(B(0, 1)) with χ ≥ 0 and � Rd χ(x)dx = 1, and set χε(x) = ε−dχ(x/ε) for ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We write E (X) for the space of smooth functions in X endowed with its natural Fr´echet space topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We set EP(X) = {f ∈ E (X) | P(D)f = 0} and endow it with the relative topology induced by E (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Let n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We write E (Xn) for the space of smooth functions in Xn endowed with its natural Fr´echet space topology, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='e, the one induced by the sequence of norms (∥ · ∥n,N)N∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We define EP(Xn) = {f ∈ E (Xn) | P(D)f = 0} 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' DEBROUWERE AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' KALMES and endow it with the relative topology induced by E (Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Since EP(Xn) is closed in E (Xn), it is a Fr´echet space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' For N ∈ N we set Un,N = {f ∈ EP(Xn) | ∥f∥n,N ≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Note that � 1 N+1Un,N � N∈N is a decreasing fundamental sequence of absolutely convex neighborhoods of 0 in E (Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Consider the projective spectrum (EP(Xn), ̺n n+1)n∈N with ̺n n+1 the restriction map from EP(Xn+1) to EP(Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Then, EP(X) = Proj(EP(Xn), ̺n n+1)n∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' By [3, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1(i)] (see also [26, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='4]), P(D) : E (X) → E (X) is surjective if and only if Proj1(EP(Xn), ̺n n+1)n∈N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Hence, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1 and a simple rescaling argument yield that P(D) : E (X) → E (X) is surjective and EP(X) satisfies (Ω) if and only if ∀n, N ∈ N ∃m ≥ n, M ≥ N ∃k ≥ m, K ≥ M ∃s, C > 0 ∀ε ∈ (0, 1) : Um,M ⊆ εUn,N + C εsUk,K, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1) where we did not write the restriction maps explicitly, as we shall not do in the sequel either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We are ready to show Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Sufficiency of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' It suffices to show (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Let n, N ∈ N be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Choose �m, �N according to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1) for n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Set m = �m + 1 and M = N + �N + deg P + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Let k ≥ m, K ≥ M be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Choose ψ ∈ D(Xm) such that ψ = 1 in a neighborhood of X �m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Pick ε0 ∈ (0, 1] such that ψ = 1 on X �m + B(0, ε0), supp ψ + B(0, ε0) ⊆ Xm, Xn + B(0, ε0) ⊆ Xn+1, Xk + B(0, ε0) ⊆ Xk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Cover the compact set Xk\\X �m by finitely many balls B(ξj, ε0), j ∈ J, with ξj ∈ X\\X �m, and choose ϕj ∈ D(B(ξj, ε0)), j ∈ J, such that � j∈J ϕj = 1 in a neighborhood of Xk\\X �m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' As J is finite, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1) for k + 1 implies that there are �K ∈ N, �s, �C > 0 such that for all ε ∈ (0, ε0) there exist Eξj,ε ∈ D′(Rd), j ∈ J, with P(D)Eξj,ε = δξj in Xk+1 such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='2) ∥Eξj,ε∥∗ n+1, ˜ N ≤ ε and ∥Eξj,ε∥∗ k+1, � K ≤ �C ε�s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Let f ∈ EP(Xm) be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' For ε ∈ (0, ε0) we define fε = (ψf) ∗ χε ∈ EP(X �m) and hε = � j∈J Eξj,ε ∗ δ−ξj ∗ (ϕjP(D)fε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Since δ−ξj ∗ (ϕjP(D)fε) = (ϕjP(D)fε)(· + ξj) ∈ D(B(0, ε0)), j ∈ J, it holds that P(D)hε = � j∈J ϕjP(D)fε in a neighborhood of Xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' As � j∈J ϕj = 1 in a neighborhood of Xk\\X �m and fε ∈ EP(X �m), we obtain that P(D)hε = P(D)fε in a neighborhood of Xk and thus hε ∈ EP(X �m) and fε − hε ∈ EP(Xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We decompose f as follows f = (f − fε + hε) + (fε − hε) ∈ EP(Xn) + EP(Xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' LINEAR TOPOLOGICAL INVARIANTS BY SHIFTED FUNDAMENTAL SOLUTIONS 7 We claim that there are s, Ci > 0, i = 1, 2, 3, 4, such that for all f ∈ EP(Xm) and ε ∈ (0, ε0) ∥f − fε∥n,N ≤ C1ε∥f∥m,M, ∥hε∥n,N ≤ C2ε∥f∥m,M, ∥fε∥k,K ≤ C3 εs ∥f∥m,M, ∥hε∥k,K ≤ C4 εs ∥f∥m,M, which implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Let f ∈ EP(Xm) and ε ∈ (0, ε0) be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' By the mean value theorem, we find that ∥f − fε∥n,N ≤ ε √ d∥f∥n+1,N+1 ≤ ε √ d∥f∥m,M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Furthermore, it holds that ∥fε∥k,K ≤ ∥χ∥L1,K εK ∥ψf∥L∞ ≤ ∥χ∥L1,K∥ψ∥L∞ εK ∥f∥m,M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' By the first inequality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='2), we obtain that ∥hε∥n,N ≤ � j∈J ∥Eξj,ε∥∗ n+1, � N∥(ϕjP(D)fε)(· + ξj)∥L∞,N+ � N ≤ ε � j∈J ∥ϕj((P(D)(ψf)) ∗ χε)∥L∞,N+ � N ≤ C′ 2ε∥P(D)(ψf)∥L∞,N+ � N ≤ C2ε∥f∥m,N+ � N+deg P ≤ C2ε∥f∥m,M, for some C′ 2, C2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Similarly, by the second inequality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='2), we find that ∥hε∥k,K ≤ � j∈J ∥Eξj,ε∥∗ k+1, � K∥(ϕjP(D)fε)(· + ξj)∥L∞,K+ � K ≤ �C ε�s � j∈J ∥ϕj((P(D)(ψf)) ∗ χε)∥L∞,K+ � K ≤ C′ 4 ε�s ∥P(D)(ψf)∥L∞∥χε∥L1,K+ � K ≤ C4 ε�s+K+ � K ∥f∥m,deg P ≤ C4 ε�s+K+ � K ∥f∥m,M, for some C′ 4, C4 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='This proves the claim with s = �s + K + �K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' □ Necessity of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' As explained above, condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Let F ∈ D′(Rd) be a fundamental solution for P(D) of finite order q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Let n ∈ N be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Choose m, � M ∈ N according to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1) for n and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Set N = q + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Let k ≥ m and ξ ∈ X\\Xm be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Set K = q + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1) for k + 1 and 0 implies that there are �C, �s > 0 such that for all δ ∈ (0, 1) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='3) Um,� M ⊆ δUn,0 + �C δ�sUk+1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' 8 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' DEBROUWERE AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' KALMES Let ε0 ∈ (0, 1] be such that B(ξ, ε0) ⊆ X\\Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Set Fξ = F ∗ δξ ∈ D′(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' For all ε ∈ (0, ε0) it holds that Fξ ∗ χε ∈ EP(Xm) and ∥Fξ ∗ χε∥m,� M ≤ C′ εd+� M+q with C′ = ∥Fξ∥∗ Xm+B(0,ε0),q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Hence, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='3) with δ = εd+� M+q+1 implies that Fξ ∗ χε ∈ C′ εd+� M+q Um,� M ⊆ C′εUn,0 + C′ �C εs Uk+1,0, with s = d + � M + q + �s(d + � M + q + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Let fξ,ε ∈ C′εUn,0 and hξ,ε ∈ C′ �Cε−sUk+1,0 be such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='4) Fξ ∗ χε = fξ,ε + hξ,ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Choose ψ ∈ D(Xk+1) such that ψ = 1 in a neighborhood of Xk and define Eξ,ε = Fξ − ψhξ,ε ∈ D′(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Then, P(D)Eξ,ε = δξ in Xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Moreover, for all ε ∈ (0, ε0) it holds that ∥Eξ,ε∥∗ n,q+1 ≤ ∥Fξ − Fξ ∗ χε∥∗ n,q+1 + ∥Fξ ∗ χε − ψhξ,ε∥∗ n,q+1 ≤ ∥Fξ∥∗ Xn+B(0,ε0),q √ dε + ∥fξ,ε∥n,0 ≤ (∥Fξ∥∗ Xn+B(0,ε0),q √ d + C′)ε, where we used the mean value theorem, and ∥Eξ,ε∥∗ k,q+1 ≤ ∥Fξ∥∗ k,q+1 + ∥ψhξ,ε∥∗ k,q+1 ≤ ∥Fξ∥∗ k,q+1 + |Xk|∥hξ,ε∥k,0 ≤ ∥Fξ∥∗ k,q+1 + C′ �C|Xk| εs , where |Xk| denotes the Lebesgue measure of Xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1(b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We write D′(X) for the space of distribu- tions in X endowed with its strong dual topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We set D′ P(X) = {f ∈ D′(X) | P(D)f = 0} and endow it with the relative topology induced by D′(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' In [27, Theorem (5)] it is shown that the mapping P(D) : D′(X) → D′(X) is surjective if and only if ∀ n ∈ N ∃ m ≥ n ∀ k ≥ m, N ∈ N, ξ ∈ X\\Xm, ε ∈ (0, 1) ∃ Eξ,ε ∈ D′(Rd) ∩ CN(Xn) with P(D)Eξ,ε = δξ in Xk such that ∥Eξ,ε∥Xn,N ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='5) Let n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We endow the space DXn of smooth functions with support in Xn with the relative topology induced by E (Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We write D′(Xn) for the strong dual of DXn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' LINEAR TOPOLOGICAL INVARIANTS BY SHIFTED FUNDAMENTAL SOLUTIONS 9 Then, D′(Xn) is a (DFS)-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We define D′ P(Xn) = {f ∈ D′(Xn) | P(D)f = 0} and endow it with the relative topology induced by D′(Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Since D′ P(Xn) is closed in D′(Xn), it is a (DFS)-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' For N ∈ N we set Bn,N = {f ∈ D′ P(Xn) | ∥f∥∗ n,N ≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Note that (NBn,N)N∈N is an increasing fundamental sequence of absolutely convex bounded sets in D′ P(Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Consider the projective spectrum (D′ P(Xn), ̺n n+1)n∈N with ̺n n+1 the restriction map from D′ P(Xn+1) to D′ P(Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Then, D′ P(X) = Proj(D′ P(Xn), ̺n n+1)n∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' By [3, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1(ii)] (see also [26, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='5]), P(D) : D′(X) → D′(X) is surjective if and only if Proj1(D′ P(Xn), ̺n n+1)n∈N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' The latter condition implies that (D′ P(Xn), ̺n n+1)n∈N is strongly reduced [26, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Hence, if P(D) : D′(X) → D′(X) is surjective, D′ P(X) satisfies (PΩ), respec- tively, (PΩ) if and only if (D′ P(Xn), ̺n n+1)n∈N does so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' A simple rescaling argument yields that (D′ P(Xn), ̺n n+1)n∈N satisfies (PΩ) if and only if ∀n ∈ N ∃m ≥ n ∀k ≥ m ∃N ∈ N ∀M ∈ N ∃K ∈ N, s, C > 0 ∀ε ∈ (0, 1) : Bm,M ⊆ εBn,N + C εsBk,K, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='6) where, as before, we did not write the restriction maps explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Similarly, the spec- trum (D′ P(Xn), ̺n n+1)n∈N satisfies (PΩ) if and only if (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='6) with “∃K ∈ N, s, C > 0” replaced by “∀s > 0 ∃K ∈ N, C > 0” holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We now show Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Sufficiency of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Clearly, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='2) implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='5) and thus that P(D) : D′(X) → D′(X) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Hence, by the above discussion, it suffices to show (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Let n ∈ N be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Choose m according to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='2) for n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Let k ≥ m be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Set N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Let M ∈ N be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Pick ε0 ∈ (0, 1] such that Xn + B(0, ε0) ⊆ Xn+1 and Xk + B(0, ε0) ⊆ Xk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Cover the compact set Xk\\Xm by finitely many balls B(ξj, ε0), j ∈ J, with ξj ∈ X\\Xm, and choose ϕj ∈ D(B(ξj, ε0)), j ∈ J, such that � j∈J ϕj = 1 in a neighborhood of Xk\\Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' As J is finite, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='2) for k+1 and M +deg P implies that there are �K ∈ N, �s, �C > 0 such that for all ε ∈ (0, ε0) there exist Eξj,ε ∈ D′(Rd) ∩ CM+deg P(Xn+1), j ∈ J, with P(D)Eξj,ε = δξj in Xk+1 such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='7) ∥Eξj,ε∥Xn+1,M+deg P ≤ ε and ∥Eξj,ε∥∗ k+1, � K ≤ �C ε�s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' 10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' DEBROUWERE AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' KALMES Pick ψ ∈ D(Xm) with ψ = 1 in a neighborhood of Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Let f ∈ D′ P(Xm) with ∥f∥∗ m,M < ∞ be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' For ε ∈ (0, ε0) we define hε = � j∈J Eξj,ε ∗ δ−ξj ∗ (ϕjP(D)(ψf)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' By the same reasoning as in the proof of part (a) it follows that hε ∈ D′ P(Xn) and ψf − hε ∈ D′ P(Xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Furthermore, as Eξj,ε ∈ D′(Rd) ∩ CM+deg P(Xn+1) and the distributions δ−ξj ∗ (ϕjP(D)(ψf)) = ϕjP(D)(ψf)(· + ξj), j ∈ J, have order at most M + deg P and are supported in B(0, ε0), it holds that hε ∈ C(Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We decompose f as follows in Xn f = ψf = hε + (ψf − hε) ∈ (D′ P(Xn) ∩ C(Xn)) + D′ P(Xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We claim that there are K ∈ N, s, C1, C2 > 0 such that for all f ∈ D′ P(Xm) with ∥f∥∗ m,M < ∞ and ε ∈ (0, ε0) ∥hε∥∗ n,0 ≤ C1ε∥f∥∗ m,M, ∥ψf − hε∥∗ k,K ≤ C2 εs ∥f∥∗ m,M, which implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Let f ∈ D′ P(Xm) with ∥f∥∗ m,M < ∞ and ε ∈ (0, ε0) be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Choose ρ ∈ D(Xm) with ρ = 1 in a neighborhood of supp ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' The first inequality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='7) implies that ∥hε∥∗ n,0 ≤ |Xn|∥hε∥n,0 ≤ |Xn| � j∈J ∥P(D)(ψf)∥∗ m,M+deg P sup x∈Xn ∥(ϕjρ)Eξj,ε(x + ξj − ·)∥L∞,M+deg P ≤ C1∥f∥∗ m,M∥Eξj,ε∥n+1,M+deg P ≤ C1ε∥f∥∗ m,M for some C1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Next, by the second inequality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='7), we obtain that for all ϕ ∈ DXk |⟨hε, ϕ⟩| ≤ � j∈J |⟨Eξj,ε ∗ δ−ξj ∗ (ϕjP(D)(ψf)), ϕ⟩| = � j∈J |⟨Eξj,ε, (δ−ξj ∗ (ϕjP(D)(ψf)))∨ ∗ ϕ⟩| ≤ � j∈J ∥Eξj,ε∥∗ k+1, � K∥(δ−ξj ∗ (ϕjP(D)(ψf)))∨ ∗ ϕ∥L∞, � K ≤ �C ε�s � j∈J ∥P(D)(ψf)∥∗ m,M+deg P sup x∈Rd ∥(ϕjρ)ϕ(· − ξj − x)∥L∞, � K+M+deg P ≤ C′ 2 ε�s ∥f∥∗ m,M∥ϕ∥L∞, � K+M+deg P, for some C′ 2 > 0, whence ∥hε∥∗ k, � K+M+deg P ≤ C′ 2 ε�s ∥f∥∗ m,M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' LINEAR TOPOLOGICAL INVARIANTS BY SHIFTED FUNDAMENTAL SOLUTIONS 11 Furthermore, ∥ψf∥∗ k, � K+M+deg P ≤ C′′ 2∥f∥∗ m,M, for some C′′ 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Therefore, ∥ψf − hε∥∗ k, � K+M+deg P ≤ C′′ 2 + C′ 2 ε�s ∥f∥∗ m,M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' This shows the claim with K = �K + M + deg P and s = �s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' □ Necessity of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' As explained above, conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='6) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Let n ∈ N be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Choose �m according to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='6) for n + 1 and m according to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='5) for �m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Let k ≥ m, N ∈ N, and ξ ∈ X\\Xm be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='5) for k and N + 1 implies that there exist Fξ ∈ D′(Rd) ∩ CN+1(X �m) with P(D)Fξ = δξ in Xk such that ∥Fξ∥ �m,N+1 ≤ min{1, 1/|X �m|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='6) for k + 1, we obtain that there are �N, �K ∈ N, �s, �C > 0 such that for all δ ∈ (0, 1) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='8) B �m,0 ⊆ δBn+1, � N + �C δ�sBk+1, � K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Note that Fξ ∈ D′ P(X �m) and ∥Fξ∥∗ �m,0 ≤ |X �m|∥Fξ∥ �m,0 ≤ 1, whence Fξ ∈ B �m,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' There- fore, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='8) yields that for all δ ∈ (0, 1) there are Gξ,δ ∈ δBn+1, � N and Hξ,δ ∈ �Cδ−�sBk+1, � K such that Fξ = Gξ,δ + Hξ,δ in Xn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Let ψ ∈ D(Xk+1) be such that ψ = 1 on a neighborhood of Xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Choose ε0 ∈ (0, 1] such that ψ = 1 on Xk + B(0, ε0) and Xn + B(0, ε0) ⊆ Xn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' For δ ∈ (0, 1) and ε ∈ (0, ε0) we define Eξ,ε,δ = Fξ − (ψHξ,δ) ∗ χε ∈ D′(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Since Hξ,δ ∈ D′ P(Xk+1), we have that (ψHξ,δ) ∗ χε ∈ EP(Xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' This implies that Eξ,ε,δ ∈ CN(Xn) and P(D)Eξ,ε,δ = δξ in Xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' As ψ = 1 on Xn+1, it holds that (ψHξ,δ) ∗ χε = Hξ,δ ∗ χε on Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Hence, we obtain that ∥Eξ,ε,δ∥n,N ≤ ∥Fξ − Fξ ∗ χε∥n,N + ∥Fξ ∗ χε − Hξ,δ ∗ χε∥n,N ≤ √ dε + ∥Gξ,δ ∗ χε∥n,N ≤ √ dε + ∥χ∥L∞,N+ � N δ εN+ � N+d, where we used the mean value theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Let r ∈ N be the order of Fξ in Xk and set K = max{r, �K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Then, ∥Eξ,ε,δ∥∗ k,K ≤ ∥Fξ∥∗ k,r + ∥(ψHξ,δ) ∗ χε∥∗ k, � K ≤ ∥Fξ∥∗ k,r + C′ δ�s , for some C′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' For ε ∈ (0, ε0) we set δε = εN+ � N+d+1 and Eξ,ε = Eξ,ε,δε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We obtain that Eξ,ε ∈ CN(Xn) with P(D)Eξ,ε = δξ in Xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Furthermore, there are C1, C2 > 0 such that for all ε ∈ (0, ε0) ∥Eξ,ε,δ∥n,N ≤ C1ε and ∥Eξ,ε,δ∥∗ k,K ≤ C2 εs with s = �s(N + � N + d + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' □ 12 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' DEBROUWERE AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' KALMES Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1(c) can be shown in the same way as Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1(b) (see in particular the values of s in terms of �s in the above proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We leave the details to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' The condition (Ω) for EP(X) if X is convex In this final section we use Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1(a) to prove that P(D) : E (X) → E (X) is surjective and EP(X) satisfies (Ω) for any non-zero differential operator P(D) and any open convex set X ⊆ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' To this end, we show that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1) holds for any exhaustion by relatively compact open convex subsets (Xn)n∈N of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' The latter is a consequence of the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Let P ∈ C[ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' , ξd]\\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Let K ⊆ Rd be compact and convex, and let ξ ∈ Rd be such that ξ /∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' For all ε ∈ (0, 1) there exists Eξ,ε ∈ D′(Rd) with P(D)Eξ,ε = δξ in Rd such that ∥Eξ,ε∥∗ K,d+1 ≤ ε and for every L ⊂ Rd compact and convex there are s, C > 0 such that ∥Eξ,ε∥∗ L,d+1 ≤ C εs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' The rest of this section is devoted to the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1, which is based on a construction of fundamental solutions due to H¨ormander [10, proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We need some preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' For Q ∈ C[ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' , ξd] we define �Q(ζ) = � � α∈Nd |Q(α)(ζ)|2 �1/2 , ζ ∈ Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Let m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We denote by Pol◦(m) the finite-dimensional vector space of non-zero polynomials in d variables of degree at most m with the origin removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' By [10, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='11 and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='12] there exists a non-negative Φ ∈ C∞(Pol◦(m)×Cd) such that (i) For all Q ∈ Pol◦(m) it holds that Φ(Q, ζ) = 0 if |ζ| > 1 and � Cd Φ(Q, ζ)dζ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' (ii) For all entire functions F on Cd and Q ∈ Pol◦(m) it holds that � Cd F(ζ)Φ(Q, ζ)dζ = F(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' (iii) There is A > 0 such that for all Q ∈ Pol◦(m) and ζ ∈ Cd with Φ(Q, ζ) ̸= 0 it holds that �Q(0) ≤ A|Q(ζ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Let K ⊆ Rd be compact and convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' As customary, we define the supporting function of K as HK(η) = sup x∈K η · x, η ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' LINEAR TOPOLOGICAL INVARIANTS BY SHIFTED FUNDAMENTAL SOLUTIONS 13 Note that HK is subadditive and positive homogeneous of degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Furthermore, it holds that [10, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='2] (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1) K = {x ∈ Rd | η · x ≤ HK(η), ∀η ∈ Rd}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We define the Fourier transform of ϕ ∈ D(Rd) as �ϕ(ζ) = � Rd ϕ(x)e−iζ·xdx, ζ ∈ Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Then, �ϕ is an entire function on Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' For all N ∈ N there is C > 0 such that for all ϕ ∈ D(Rd) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='2) |�ϕ(ζ)| ≤ C∥ϕ∥L1,N eHch supp ϕ(Imζ) (2 + |ζ|)N , ζ ∈ Cd, where ch supp ϕ denotes the convex hull of supp ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We are ready to show Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We may assume without loss of generality that ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Since 0 /∈ K, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='1) implies that there is η ∈ Rd such that HK(−η) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' For t > 0 and σ ∈ Rd we define Pt,σ = P(σ +itη + · ) ∈ Pol◦(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Note that there is c > 0 such for all t > 0 and σ ∈ Rd (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='3) � Pt,σ(0) = �P(σ + itη) ≥ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Let Φ be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' We define Ft ∈ D′(Rd) via (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' [10, proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='10]) ⟨Ft, ϕ⟩ = 1 (2π)d � Rd � Cd �ϕ(−σ − itη − ζ) P(σ + itη + ζ) Φ(Pt,σ, ζ)dζdσ, ϕ ∈ D(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Let L be an arbitrary compact convex subset of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' By properties (i) and (iii) of Φ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='2) (with N = d + 1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='3) we have that for all ϕ ∈ DL |⟨Ft, ϕ⟩| ≤ 1 (2π)d � Rd � |ζ|≤1 |�ϕ(−σ − itη − ζ)| |Pt,��(ζ)| Φ(Pt,σ, ζ)dζdσ ≤ AC∥ϕ∥L1,d+1 (2π)d � Rd � |ζ|≤1 eHL(−tη−Im ζ) (2 + |σ + itη + ζ|)d+1� Pt,σ(0) Φ(Pt,σ, ζ)dζdσ ≤ AC∥ϕ∥L1,d+1 (2π)dc � Rd � |ζ|≤1 etHL(−η)eHL(− Im ζ) (1 + |σ|)d+1 Φ(Pt,σ, ζ)dζdσ ≤ C′ L∥ϕ∥L∞,d+1etHL(−η), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='4) where C′ L = AC|L| (2π)dc max |ζ|≤1 eHL(− Im ζ) � Rd 1 (1 + |σ|)d+1dσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' In particular, Ft is a well-defined distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Property (ii) of Φ and Cauchy’s integral formula yield that for all ϕ ∈ D(Rd) ⟨P(D)Ft, ϕ⟩ = ⟨Ft, P(−D)ϕ⟩ = 1 (2π)d � Rd � Cd �ϕ(−σ − itη − ζ)Φ(Pt,σ, ζ)dζdσ = 1 (2π)d � Rd �ϕ(−σ − itη)dσ 14 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' DEBROUWERE AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' KALMES = 1 (2π)d � Rd �ϕ(σ)dσ = ϕ(0) and thus P(D)Ft = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' For ε ∈ (0, 1) we set tε = log ε/HK(−η) > 0 and E0,ε = Ftε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Then, P(D)E0,ε = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content='4), we obtain that for all ε ∈ (0, 1) ∥E0,ε∥∗ K,d+1 ≤ C′ Kε and, for any L ⊆ Rd compact and convex, ∥E0,ε∥∗ L,d+1 ≤ C′ L εs , with s = |HL(−η)/HK(−η)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' This gives the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' □ References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Bonet, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfvQHn/content/2301.02617v1.pdf'} +page_content=' Doma´nski, Parameter dependence of solutions of differential equations on spaces of distributions and the splitting of short exact 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