diff --git "a/D9E1T4oBgHgl3EQfEQOb/content/tmp_files/2301.02888v1.pdf.txt" "b/D9E1T4oBgHgl3EQfEQOb/content/tmp_files/2301.02888v1.pdf.txt" new file mode 100644--- /dev/null +++ "b/D9E1T4oBgHgl3EQfEQOb/content/tmp_files/2301.02888v1.pdf.txt" @@ -0,0 +1,2253 @@ +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL +APPROXIMATION +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +Abstract. We strengthen the classical approximation theorems of Weierstrass, Runge and +Mergelyan by showing the polynomial and rational approximants can be taken to have a +simple geometric structure. In particular, when approximating a function f on a compact +set K, all the critical points of our approximants lie close to K, and all the critical values +lie close to f(K). Our proofs rely on extensions of (1) the quasiconformal folding method of +the first author, and (2) a theorem of Carath´eodory on approximation of bounded analytic +functions by finite Blaschke products. +1. Introduction +The following is Runge’s classical theorem on polynomial approximation. +Theorem 1.1. [Run85] Let f be a function analytic on a neighborhood of a compact set +K ⊂ C, and suppose C \ K is connected. For all ε > 0, there exists a polynomial p so that +||f − p||K := sup +z∈K +|f(z) − p(z)| < ε. +This famous result does not say much about what the polynomial approximant p looks +like off the compact set. For various applications, it would be useful to understand the global +behavior of p and, in particular, the location of the critical points and values of p. To this +end, we state our first result (Theorem A below) after introducing the following notation. +Notation 1.2. For any compact set K ⊂ C we denote by fill(K) the union of K with all +bounded components of C \ K. We say K is full if C \ K is connected. We let CP(f) denote +the set of critical points of an analytic function f, and let CV(f) := f(CP(f)) denote its +critical values. A domain in �C is an open, connected subset of �C. +Theorem A. (Polynomial Runge+) +Let K ⊂ C be compact and full, D a domain +containing K, and suppose f is a function analytic in a neighborhood of K. Then for all +ε > 0, there exists a polynomial p so that ||p − f||K < ε and: +(1) CP(p) ⊂ D, +(2) CV(f|K) ⊂ CV(p) ⊂ fill{z : d(z, f(K)) ≤ ε}. +2020 Mathematics Subject Classification. Primary: 30C10, 30C62, 30E10, Secondary: 41A20. +Key words and phrases. uniform approximation, polynomials, rational functions, Blaschke products, +Runge’s Theorem, Weierstrass’s Theorem, Mergelyan’s Theorem. +The first author is partially supported by NSF Grant DMS 1906259. +1 +arXiv:2301.02888v1 [math.CV] 7 Jan 2023 + +2 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +Analogous improvements of the polynomial approximation theorems of Mergelyan and +Weierstrass will be stated and proved in Section 9 (see Theorem 9.8 and Corollary 9.9). +When K is not full, uniform approximation by polynomials is not always possible, and so +we turn to rational approximation. We denote the Hausdorff distance between two sets X, +Y , by dH(X, Y ). +Theorem B. (Rational Runge+) Let K ⊂ C be compact, D a domain containing K, f +a function analytic in a neighborhood of K, and suppose P ⊂ �C \ K contains exactly one +point from each component of �C \ K. Then there exists P ′ ⊂ P so that for all ε > 0, there +is a rational function r so that ||r − f||K < ε and: +(1) dH(r−1(∞), P ′) < ε and |r−1(∞)| = |P ′|, +(2) CP(r) ⊂ D, +(3) CV(f|K) ⊂ CV(r) ⊂ fill{z : d(z, f(K)) ≤ ε}. +The behavior of p off K is of particular interest in applications, such as in complex dy- +namics where approximation results have been used to prove the existence of various dy- +namical behaviors for entire functions (see, for example, [EL87], [BT21], [MRW21], [ERS22], +[MRW22], [BEF+22]). However, not understanding the critical points and values of p means +it has not been known whether these behaviors can occur within restricted classes of en- +tire functions, such as the well studied Speiser or Eremenko-Lyubich classes (see the survey +[Six18]). +Our approach is based on two ideas. The first is to show that on a compact subset K ⊂ C +of a finitely connected domain Ω ⊂ C, any bounded analytic function can be approximated +uniformly by an analytic function B : Ω → C having the property that |B| is constant on +each component of ∂Ω. This extends a classical theorem of Carath´eodory [Car54] concerning +finite Blaschke products on the unit disk to more general regions, and may be of independent +interest. +The second idea is an extension of quasiconformal folding, a type of quasiconformal surgery +introduced in [Bis15], to extend the (generalized) Blaschke product B from Ω to a quasireg- +ular mapping g : �C → �C with specified poles. The map g may be taken close to holomorphic +in a suitable sense, and so the Measurable Riemann Mapping Theorem (MRMT for brevity) +will imply there is a quasiconformal mapping φ so that g ◦ φ−1 is the desired polynomial or +rational approximant. +This approach yields not only information on the critical points and values of the approx- +imants as in Theorems A and B, but more broadly a detailed description of the geometric +structure of these approximants. +We end the introduction by describing this geometric +structure in a few cases. First we introduce some more notation. +Notation 1.3. Let D ⊂ �C be a simply connected domain so that ∞ ̸∈ ∂D. +We let +ψD : E → D denote a Riemann mapping, where E = D is D is bounded and E = D∗ := �C\D +if D is unbounded, in which case we specify ψD(∞) = ∞. + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +3 +First consider the case when K is full and connected, and f is holomorphic in a neighbor- +hood of K satisfying ||f||K < 1. Let Ω, Ω′ be analytic Jordan domains containing K, f(K), +respectively, so that f is holomorphic in Ω. Then the mapping +F := ψ−1 +Ω′ ◦ f ◦ ψΩ : D → D +is holomorphic, and by Carath´eodory’s theorem for the disk (see Theorem 2.1), there is +a finite Blaschke product b : D → D that approximates F on the compact set ψ−1 +Ω (K). +Therefore +B := ψΩ′ ◦ b ◦ ψ−1 +Ω : Ω → Ω′ +is a holomorphic function that approximates f on K, and moreover B restricts to an analytic, +finite-to-1 map of Γ := ∂Ω onto Γ′ := ∂Ω′. +In this paper, we will show that B can be approximated on Ω by a polynomial p so that +p−1(Γ′) is an approximation of Γ. More precisely, p−1(Γ′) is connected, and consists of a +finite union of Jordan curves {γj}n +0 bounding pairwise disjoint Jordan domains {Ωj}n +0 (see +Figure 1): the {Ωj}n +0 are precisely the connected components of p−1(Ω′). There is one “large” +component Ω0 that approximates Ω in the Hausdorff metric. The other components {Ωj}n +1 +can be made as small as we wish and to lie in any given neighborhood of ∂K. Moreover, the +collection {Ωj}n +0 forms a tree structure with any two boundaries ∂Ωj, ∂Ωk either disjoint or +intersecting at a single point, and with Ω0 as the “root” of the tree as in Figure 1. Let Ω∞ +denote the unbounded component of C \ p−1(Γ′), so that +(1.1) +C \ p−1(Γ′) = Ω0 ⊔ +� +⊔n +j=1Ωj +� +⊔ Ω∞. +Recalling Notation 1.3, the polynomial p has the following simple structure with respect to +the domains in (1.1). +(1) p(Ω0) = Ω′ and ψ−1 +Ω′ ◦ p ◦ ψΩ0 is a finite Blaschke product. +(2) p(Ωj) = Ω′ and p is conformal on Ωj for 1 ≤ j ≤ n. +(3) p(Ω∞) = C \ Ω′ and p = ψC\Ω′ ◦ (z �→ zm) ◦ ψ−1 +Ω∞ on Ω∞ for m = deg(p|Ω0) + n. +In other words, up to conformal changes of coordinates, p is simply a Blaschke product in +Ω0, a conformal map in each Ωj, 1 ≤ j ≤ n, and a power map z �→ zm in Ω∞. The only finite +critical points of p are either in Ω0, or at a point where two of the curves (γj)n +j=1 intersect, +in which case the corresponding critical value lies on ∂Ω′. +Next suppose K is connected, but C \ K has more than one component. In this case, in +order to prove Theorem B, we will need to let Ω be a multiply connected analytic domain +containing K, and Ω′ an analytic Jordan domain containing f(K). We cannot proceed as in +the case C\K is connected, however, without a multiply connected version of Carath´eodory’s +Theorem for the disk (Theorem 2.1). The usual proof of Carath´eodory’s Theorem is based +on power series, and it does not extend to the multiply connected setting. However, there +is an alternate proof based on potential theory that does extend. We now briefly sketch the +idea. + +4 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +K +p +Ω∞ +f(K) +Γ′ +Ω0 +{Ωj}n +1 +Figure 1. This figure illustrates the geometry of the approximant p in Theorem A +when K is connected. The notation is explained in the text. Both domain and co- +domain are colored so that regions with the same color correspond to one another +under p. +Suppose f is holomorphic in a simply connected domain D. To simplify matters, suppose +f(D) is compactly contained in D \ {0}. Then u := − log |f| is a positive harmonic function +on D, and u is also bounded and bounded away from zero. Thus u is the Poisson integral +of its boundary values. +The Poisson kernel Px(z) associated to any point x ∈ ∂D is a +limit of normalized Green’s functions on D: Px(z) ≈ G(z, wn)/G(0, wn) with wn → x. +Approximating the Poisson integral by a Riemann sum gives an approximation of u on +any compact set K ⊂ D by a sum of Green’s functions H(z) := � +j G(z, wj) with poles +distributed on {z : d(z, ∂D) = δ}, and with δ as small as we wish. Figure 2 shows the +function u(x, y) = 1 +2 + xy being approximated by a sum of 25 Green’s functions on D = D. +Since D is simply connected, H has a well defined harmonic conjugate �H, and after adding +a constant to �H if necessary, +B := exp(−H − i �H) +approximates f on K. Moreover, B satisfies: +(1) B is holomorphic on D. +(2) |B| extends continuously to a constant function on ∂D, with ||B||∂D = 1. +We call such a function B on a (not necessarily simply connected) domain D a (generalized) +finite Blaschke product on D (see Definition 2.2). When D = D this definition coincides with + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +5 +Figure 2. +On the left is the positive harmonic function u(x, y) = 1 +2 + xy +on the unit disk. On the right is a sum of 25 Green’s functions with poles +on the circle of radius .98. On D(0, 1 +2) the two functions agree to within .03. +As expected, the poles are closer together where u is large and farther apart +where u is small. +the usual definition of Blaschke product, and the above argument yields Carath´eodory’s The- +orem. In fact, the above argument yields several technical improvements of Carath´eodory’s +Theorem (see Theorem 2.6) which we will need in order to prove Theorem A. +This argument generalizes to finitely connected domains D, except that the sum of Green’s +functions H may not have a well defined harmonic conjugate (even modulo 2π). We will fix +this by adding a small harmonic function h that is constant on each boundary component +of D and whose periods match the periods of −H around each boundary component of +D. Exponentiating the sum of the modified function H + h with its harmonic conjugate +(now well-defined modulo 2π) then gives a (generalized) finite Blaschke product B which +approximates the given function f on the desired compact set K ⊂ D. This gives a version +of Carath´eodory’s theorem on finitely connected domains. By choosing H correctly, we can +take ∥h∥D as small as we wish, and hence |B| is constant on each connected component of +∂D, with values that can all be taken as close to 1 as desired. +Now we return to the description of our rational approximant in the case that K is +connected, but C\K has more than one component, recalling that Ω is a multiply connected +analytic domain containing K, and Ω′ is an analytic Jordan domain containing f(K). By +the multiply connected version of Caratheodory’s Theorem, there exists a (generalized) finite +Blaschke product b approximating ψ−1 +Ω′ ◦ f on K, so that B := ψΩ′ ◦ b is a holomorphic map +approximating f on K, and B restricts to an analytic, finite-to-1 mapping of each component +of Γ := ∂Ω onto ψΩ′(|z| = t) for t = 1 or t ≈ 1, where t may depend on the component of Γ. + +1.5 +0.5 ~ +0 +0.5 +-0.5 +-0.5 +0.51.6 +1.4 +1.2 ~ +0.8 +0.6 - +0.2 - +0.5 +0.5 +0.56 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +We will show B can be approximated on Ω by a rational map r so that each component of ∂Ω +can be approximated by a component of r−1 ◦ ψΩ′(|z| = t) for t as above. These components +of r−1 ◦ ψΩ′(|z| = t) bound Jordan domains which form a decomposition of the plane as in +the previously described polynomial setting, and in the interior of each such domain again r +behaves either as a (generalized) finite Blaschke product, a conformal mapping, or a power +mapping (up to conformal changes of coordinates). +Lastly, the case when K has more than one connected component is more intricate, and +we will leave the precise description to later in the paper. (Briefly, quasiconformal folding is +applied not just along the boundary of a neighborhood of K, but also along specially chosen +curves that connect different connected components of this neighborhood.) +We remark that while Theorem A strictly improves on Runge’s Theorem on polynomial +approximation, the relationship between Theorem B and Runge’s Theorem on rational ap- +proximation is more subtle. Both show existence of rational approximants, and only The- +orem B describes the critical point structure of the approximant, however the poles of the +approximant in Theorem B are specified only up to a small perturbation, whereas in Runge’s +Theorem they are specified exactly. We do not know whether it is necessary to consider per- +turbations of P ′ in Theorem B, or if the improvement r−1(∞) = P ′ is possible (a related +problem appears in [BL19], [DKM20], [BLU], where it is known no such improvement is +possible). +Acknowledgements. The authors would like to thank Malik Younsi and Oleg Ivrii for useful +discussions related to this manuscript, and Xavier Jarque for his comments on a preliminary +version of this manuscript. +2. Interior Approximation +The following classical result of Carath´eodory (referenced in the introduction) allows for +approximation by Blaschke products in simply-connected domains. +Theorem 2.1. ([Car54]) Let f : D → C be holomorphic and suppose ||f||D ≤ 1. Then there +exists a sequence of finite Blaschke products on D converging to f uniformly on compact +subsets of D. +The proof of Theorem 2.1 is elementary and may be found, for example, in Theorem I.2.1 of +[Gar81] or Theorem 5.1 of the survey [GMR17]. In order to prove Theorem A, we will need +to prove a version of Theorem 2.1 in which the Blaschke products satisfy certain boundary +regularity conditions, and in order to prove Theorem B, we will need to prove a multiply- +connected version of Theorem 2.1. These improvements are stated below in Theorem 2.6. +First we need several definitions: +Definition 2.2. Let D ⊂ C be a finitely connected domain. We say a non-constant holo- +morphic function B : D → C is a finite Blaschke product on D if |B| extends continuously +to a non-zero, constant function on each component of ∂D and ||B||D = 1. + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +7 +Remark 2.3. When D = D, the definition above corresponds with the usual definition of +finite Blaschke product. +Notation 2.4. For a finite Blaschke product B on a finitely connected domain D, we let +IB denote the connected components of ∂D \ {z : B(z) ∈ R}. In other words, IB are the +preimages (under B) of the open upper and lower half-circle components of B(∂D) ∩ H, +B(∂D) ∩ (−H). We will frequently be dealing with sequences of finite Blaschke products +(Bn)∞ +n=1 on D, in which case we abbreviate IBn by In. +Definition 2.5. We call a domain D ⊂ C an analytic domain if D is finitely connected, and +each component of ∂D is an analytic Jordan curve. +We remark that a boundary component of an analytic domain D cannot be a single point. +Theorem 2.6. Let D ⊂ C be an analytic domain, suppose K ⊂ D is compact, and let f be +a function analytic in a neighborhood of D satisfying ||f||D < 1 and {z : f(z) = 0}∩∂D = ∅. +Then there exists M < ∞ and a sequence (Bn)∞ +n=1 of finite Blaschke products on D satisfying: +(2.1) +inf +z∈∂D |Bn(z)| +n→∞ +−−−→ 1, +(2.2) +||Bn − f||K +n→∞ +−−−→ 0, +(2.3) +sup +I∈In +diam(I) +n→∞ +−−−→ 0, and +(2.4) +sup +I,J∈In +diam(I)/diam(J) < M and +(2.5) +sup +I∈In +supz∈I |B′ +n(z)| +infz∈I |B′ +n(z)| < M for all n. +Theorem 2.6 will suffice for the proofs of Theorems A and B, although we prove a slightly +stronger result in Section 12 (see Theorem 12.2). The proof of Theorem 2.6 will be delayed +until Sections 10-12, which are independent of Sections 2-9. +We now turn to applying +Theorem 2.6 to produce Blaschke approximants as described in the introduction. Given a +Jordan curve γ ⊂ C, we denote the bounded component of �C \ γ by int(γ). +Notation 2.7. We refer to Figure 3 for a summary of the following. For the remainder of +this section, we will fix a compact set K, an analytic domain D containing K, and a function +f holomorphic in a neighborhood of D satisfying ||f||D < 1. Fix ε > 0. We assume that +(2.6) +d(z, K) < ε/2 and d(f(z), f(K)) < ε/2 for every z ∈ ∂D. +Definition 2.8. We let γ be an analytic Jordan curve surrounding f(D) such that +(2.7) +dist(w, f(K)) < ε for every w ∈ γ, +and let Ψ : D → int(γ) denote a Riemann mapping. + +8 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +Figure 3. This figure illustrates Notations 2.4, 2.7 and Theorem 2.6. The +vertices pictured on ∂D are B−1(±1), and the components IB of Notation 2.4 +are the edges along ∂D connecting these vertices. +Remark 2.9. By enlarging D slightly we may ensure {z : Ψ−1 ◦ f(z) = 0} ∩ ∂D = ∅, so +that Theorem 2.6 applies to the triple D, K, Ψ−1 ◦ f to produce M < ∞ and a sequence of +finite Blaschke products (Bn)∞ +n=1 on D satisfying the conclusions of Theorem 2.6. +Note that in Remark 2.9 we are applying Theorem 2.6 to Ψ−1 ◦ f (and not f). This will +ensure that the critical values of the approximant Ψ ◦ Bn ≈ f are close to fill(f(K)) as +needed for Theorem B. We will now quasiconformally perturb the sequence (Bn) so as to +ensure that we can later prove the conclusion CV(f|K) ⊂ CV(r) of Theorem B: +Theorem 2.10. We may take the sequence (Bn)∞ +n=1 of Remark 2.9 so that it also satisfies: +(2.8) +CV(Ψ−1 ◦ f|K) ⊂ CV(Bn) for all large n. +Proof. Let K′ ⊂ D be a compact set satisfying K ⊂ int(K′) ⊂ D. Apply Theorem 2.6 to +the triple D, K′, Ψ−1 ◦ f to obtain a sequence of Blaschke products (Bn)∞ +n=1 on D. Let +z ∈ CP(f|K). Since +(2.9) +||Bn − Ψ−1 ◦ f||K′ +n→∞ +−−−→ 0, +by Hurwitz’s Theorem there exists a sequence (wn +z )∞ +n=1 such that +B′ +n(wn +z ) = 0 and wn +z +n→∞ +−−−→ z. +Let r < 1 be so that ψ−1 ◦ f(K) ⊂ rD. Define a homeomorphism hn : D → D to be an +interpolation of +(2.10) +hn(Bn(wn +z )) := Ψ−1 ◦ f(z) for all z ∈ CP(f|K), and + +int() +f(K) +Y +Izl<1 +D +K +B ~ +ofA GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +9 +(2.11) +hn(z) = z for r ≤ |z| ≤ 1. +By (2.9), we have that +|Bn(wn +z ) − Ψ−1 ◦ f(z)| +n→∞ +−−−→ 0 for all z ∈ CP(f|K), +and hence hn may be taken to satisfy: +(2.12) +||(hn)z/(hn)z||D +n→∞ +−−−→ 0. +By the Measurable Riemann Mapping Theorem (see [Ahl06]), there is a quasiconformal +φn : D → D so that: +Bn := hn ◦ Bn ◦ φ−1 +n +is holomorphic. We normalize each φn by specifying φn(p) = p and φ′ +n(p) > 0 for some +r < |p| < 1. Note that +(2.13) +Bn(∂D) ⊂ {z : r ≤ |z| ≤ 1} for large n +by (2.1), so it follows from (2.11) that Bn is a Blaschke product on D for large n. We claim +the sequence (Bn) satisfies the conclusions of Theorem 2.6, as well as (2.8). Indeed, we have +φn(wn +z ) ∈ CP(Bn) and +Bn(φn(wn +z )) = Ψ−1 ◦ f(z). +Thus (2.8) is satisfied. By (2.12), we have that +(2.14) +||φn(z) − z||D +n→∞ +−−−→ 0, +and hence (2.2) follows. The relation (2.1) follows from (2.11), and (2.3) follows from (2.14). +To prove (2.4) and (2.5), we first note that since (hn)z = 0 in r < |z| < 1, it follows from +(2.13) that φn extends to a holomorphic function in a neighborhood U of ∂D, where U does +not depend on n. By (2.14) and the Cauchy integral formula, it follows that φ′ +n(z) +n→∞ +−−−→ 1 +uniformly for z ∈ ∂D and hence we deduce (2.4). Similarly, φ′′ +n(z) +n→∞ +−−−→ 0 uniformly for +z ∈ ∂D and so (2.5) follows. +□ +Recall from the introduction that we plan to extend the definition of the approximant +Ψ ◦ Bn ≈ f from D to all of C, where we recall Ψ was defined in Definition 2.8. To this end, +it will be useful to define the following graph structure on ∂D. +Definition 2.11. For any n ∈ N, we define a set of vertices on ∂D by Vn := (Bn|∂D)−1(R), +where each vertex v is labeled black or white according to whether Bn(v) > 0 or Bn(v) < 0, +respectively. The curve ∂D will be considered as a graph with edges defined by In (recall +from Notation 2.4 that In is precisely the collection of components of ∂D \ Vn). We will +sometimes write Dn in place of D when we wish to emphasize the dependence of the graph +∂D on n. +Definition 2.12. We define a holomorphic mapping gn in D by the formula +(2.15) +gn(z) := Ψ ◦ Bn(z). + +10 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +In Sections 3-7 we will quasiregularly extend the definition of gn to C, and then in Section +9 we apply the MRMT to produce the rational approximant of Theorem B as described in +the introduction. +Remark 2.13. Recall that in Notation 2.7, we fixed ε > 0, a compact set K contained in +an analytic domain D, and a function f holomorphic in D (we note ε, K, D, f also satisfied +extra conditions specified in Notation 2.7). The objects γ, Ψ, Bn, Vn, gn we then defined in +this section were determined by our initial choice of ε, K, D, f. In future sections, it will be +useful to think of γ, Ψ, Bn, Vn, gn as defining functions which take as input some quadruple +(ε, K, D, f) (for any ε, K, D, f as in Notation 2.7), and output whatever object we defined +in this section. For instance, Vn defines a function which takes as input any (ε, K, D, f) +as in Notation 2.7 and outputs (via Definition 2.11) a set of vertices Vn(ε, K, D, f) on ∂D. +Similarly, Bn takes as input any (ε, K, D, f) as in Notation 2.7 and outputs (via Theorem +2.10) a Blaschke product Bn(ε, K, D, f) on D. Likewise for γ, Ψ, gn. +3. Quasiconformal Folding +Given a compact set K ⊂ C and a function f holomorphic in a domain D containing +K, we showed in Section 2 how to approximate f by a holomorphic function gn defined in +D (see Definition 2.12). Moreover gn is just a Blaschke product in D. If f is a function +holomorphic in an arbitrary analytic neighborhood U (where U need not be connected) of a +compact set K, then one can apply the results of Section 2 to each component of U which +intersects K (this is done precisely in Definition 4.1): this yields a holomorphic approximant +of f defined in a finite union of domains, so that the approximant is just a finite Blaschke +product on each domain (recall Definition 2.2). In Sections 3-7, we will build the apparatus +necessary to systematically extend this holomorphic approximant to a quasiregular function +of C which is holomorphic outside a small set. +It was convenient to assume in Notation 2.7 that the compact set K was covered by a +single domain D, however we now begin to work more generally: +Remark 3.1. We refer to Figure 6 for a summary of the following. Throughout Sections +3-7, we will fix ε > 0, a compact set K ⊂ C, a domain D containing K, a disjoint collection +of analytic domains (Di)k +i=1 such that K ⊂ U := ∪iDi ⊂ D, and a function f holomorphic in +a neighborhood of U satisfying ||f||U < 1. We assume that the following analog of Equation +(2.6) holds +(3.1) +d(z, K ∩ Di) < ε/2 and d(f(z), f(K ∩ Di)) < ε/2 for all z ∈ ∂Di and 1 ≤ i ≤ k. +Applying the methods of the previous section to each component Di of U, we can define a +sequence of finite Blaschke products (Bn)∞ +n=1 on each Di (see Remark 2.13). We will let Bn +denote the corresponding function defined on U. In particular, (Bn)∞ +n=1 gives the following +definition of vertices on the boundary of U := ∪iDi (see Definition 2.11 and Remark 2.13). + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +11 +Figure 4. Illustrated is the Definition of the domain V in the proof of Propo- +sition 3.3 +Definition 3.2. For every n ∈ N, we define a set of vertices Vn on ∂U by +Vn := +k� +i=1 +Vn(ε, K ∩ Di, Di, f|Di) = +k� +i=1 +(Bn|∂Di)−1(R). +We now extend the graph structure on ∂U by connecting the different components of +U by curves {Γi}k−1 +i=1 in Proposition 3.3 below, and defining vertices along these curves in +Definition 3.4. We will need to prove a certain level of regularity for these curves and vertices +in order to ensure that the dilatations of quasiconformal adjustments we will make later do +not degenerate as n → ∞. We will denote the curves by {Γi}k−1 +i=1 , and we remark that the +curves depend on n, although we suppress this from the notation. +Proposition 3.3. For each n ∈ N, there exists a collection of disjoint, closed, analytic +Jordan arcs {Γi}k−1 +i=1 in (�C \ U) ∩ D satisfying the following properties: +(1) Each endpoint of Γi is a vertex in Vn, +(2) Each Γi meets ∂U at right angles, +(3) U ∪ (∪iΓi) is connected, and +(4) For each 1 ≤ i ≤ k − 1, the sequence (in n) of curves Γi has an analytic limit. +Proof. The set (�C \ U) ∩ D must contain at least one simply-connected region V with the +property that there are distinct i, j with both ∂V ∩∂Di and ∂V ∩∂Dj containing non-trivial +arcs (see Figure 4). By (2.3), for all sufficiently large n both ∂V ∩ ∂Di, ∂V ∩ ∂Dj contain +vertices of Vn which we denote by vi ∈ ∂Di, vj ∈ ∂Dj, respectively. Consider a conformal +map φ : D → V , and define Γ1 to be the image under φ of the hyperbolic geodesic connecting +φ−1(vi), φ−1 +i (vj) in D. +We now proceed recursively, making sure at step l we pick a V which connects two com- +ponents of U not already connected by a Γ1, ..., Γl−1, and so that V is disjoint from Γ1, ..., +Γl−1. The curves Γi satisfy conclusions (1)-(3) of the proposition. We may ensure that for +each 1 ≤ i ≤ k − 1, the sequence (in n) of curves Γi has an analytic limit by choosing vi, vj +above to converge as n → ∞. +□ + +D +D2 +V +IJ +D112 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +Figure 5. Illustrated is Definition 3.4. +Definition 3.4. Consider the vertices Vn ⊂ ∂U of Definition 3.2. We will augment Vn to +include vertices on the curves (Γi)k−1 +i=1 as follows (see Figure 5). Let Γ ∈ (Γi)k−1 +i=1 denote both +the curve as a subset of C and the arclength parameterization of the curve, and suppose Γ +connects vertices +Γ(0) = vi ∈ ∂Di, Γ(length(Γ)) = vj ∈ ∂Dj. +For k = i, j, let εk denote the minimum length of the two edges with endpoint vk in ∂Dk, +and suppose without loss of generality εj < εi. Let l be so that +εj/2 ≤ εi/2l ≤ 2εj. +We place vertices at Γ(εi/2), ..., Γ(εi/2l), and we place vertices along Γ([εi/2l, length(Γ)]) +at equidistributed points. We can label the vertices black/white along Γ so that vertices +connect only to vertices of the opposite color by adding one extra vertex at the midpoint of +the segment having vj as an endpoint, if need be. +We introduce the following notation. +Notation 3.5. Throughout Sections 3-7, we will let Ω denote a fixed (arbitrary) component +of +(3.2) +�C \ +� +U ∪ +k−1 +� +i=1 +Γi +� +, +and p ∈ Ω. Note that Ω is simply connected by Proposition 3.3(3). Denote D∗ := �C\D, and +let σ denote any conformal mapping +(3.3) +σ : D∗ → Ω +satisfying σ(∞) = p. For z ∈ Ω, we define τ(z) := σ−1(z). The map τ induces a partition of +T which we denote by Vn := τ(Vn). + +D1 +E1 +82A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +13 +|z|>1 +Ω +τ +3 +D +D2 +Γ2 +1Γ +Γ3 +D4 +D1 +Figure 6. This figure illustrates Remark 3.1 and Notation 3.5. As pictured, +U has four components (Di)4 +i=1 which are connected by curves (Γi)3 +i=1. Recall +K ⊂ U (the compact set K is not shown in the figure). +The unbounded +component Ω of (3.2) is pictured in dark grey. +The map τ : Ω → D∗ is +a conformal mapping, and sends the vertices on ∂Ω to (possibly unevenly +spaced) vertices on the unit circle. +Remark 3.6. We will sometimes write Ωn, D∗ +n in place of Ω, D∗, respectively, when we wish +to emphasize the dependence of the vertices Vn ⊂ ∂Ω, Vn ⊂ ∂D∗ on the parameter n. +Proposition 3.7. For the graph ∂Ωn, we have: +max{diam(e) : e is an edge of ∂Ωn} +n→∞ +−−−→ 0. +Proof. This follows from (2.3) and Definition 3.4. +□ +As explained in the introduction, in order to prove uniform approximation in Theorem +B, we will need to prove that our quasiregular extension is holomorphic outside a region of +small area. This will usually mean proving the following condition holds. +Definition 3.8. Suppose V ⊂ C is an analytic domain, and ∂V is a graph. Let C > 0. We +say a quasiregular mapping φ : V → φ(V ) is C-vertex-supported if +(3.4) +supp(φz) ⊂ +� +e∈∂V +{z : dist(z, e) < C · diam(e)} +(see Figure 7), where the union in (3.4) is taken over all edges e on ∂V . +It will also be useful to have the following definition. +Definition 3.9. Suppose e, f are rectifiable Jordan arcs, and h : e → f is a homeomorphism. +We say that h is length-multiplying on e if the push-forward (under h) of arc-length measure +on e coincides with the arc-length measure on f multiplied by length(f)/length(e). + +14 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +Figure 7. Shown as a black curve is part of a graph G, and in light gray the +neighborhood ∪e∈G{z : dist(z, e) < C · diam(e)} of G. +First we will adjust the conformal map τ so as to be length-multiplying along edges of +∂Ω. Recall the vertices Vn ⊂ T defined in Notation 3.5. +Proposition 3.10. For every n, there is a K-quasiconformal mapping λ : D∗ +n → D∗ +n so that: +(1) λ is C-vertex-supported for some C > 0, +(2) λ(z) = z on Vn and off of supp(λz), +(3) λ ◦ τ is length-multiplying on every component of ∂Ω \ Vn, +(4) C, K do not depend on n. +Proof. This is a consequence of Theorem 4.3 of [Bis15]. Indeed, recall τ := σ−1 and consider +the 2πi-periodic covering map +(3.5) +φ := σ ◦ exp : Hr �→ Ω. +The map φ induces a periodic partition φ−1(Vn) of ∂Hr which has bounded geometry (see +the introduction of [Bis15], or Section 2 of [BL19]) with constants independent of n by +Proposition 3.3(2) and Definition 3.4. Thus Theorem 4.3 of [Bis15] applies to produce a +2πi-periodic, C vertex-supported, and K-quasiconformal map β : Hr → Hr so that φ ◦ β is +length-multiplying on edges of Hr, and C, K are independent of n. Thus, the inverse +β−1 ◦ log ◦τ +is length-multiplying, and since exp is length-multiplying on vertical edges, the well-defined +map +λ := exp ◦β−1 ◦ log : D∗ → D∗ +satisfies the conclusions of the Proposition. +□ +The main idea in defining the quasiregular extension in Ω is to send each edge of ∂Di +to the upper or lower half of the unit circle by following λ ◦ τ with a power map z �→ zn + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +15 +Figure 8. This figure illustrates the Folding Theorem 3.13 and Notation 3.14. +The simply connected domain Ω′ +n is obtained by removing from Ω certain trees +based at the vertices along ∂Ω. +of appropriate degree. The main difficulty in this approach, however, is that the images of +different edges of ∂Di under λ ◦ τ may differ significantly in size, so that there is no single +n with z �→ zn achieving the desired behavior. The solution is to modify the domain Ω by +removing certain “decorations” from the domain Ω, so that each edge of ∂Di is sent to an +arc of roughly the same size under λ ◦ τ. This is formalized below in Theorem 3.13 (see also +Figures 8, 9), and is an application of the main technical result of [Bis15] (see Lemma 5.1). +The “decorations” are the trees in the following definition. +Definition 3.11. Let V ⊂ T be a discrete set. We call a domain W ⊂ D∗ a tree domain +rooted at V if W consists of the complement in D∗ of a collection of disjoint trees, one rooted +at each vertex of V (see the center of Figure 8). +Notation 3.12. For m ∈ N, we let +Z± +m := {z ∈ T : zm = ±1}, +Zm := Z+ +m ∪ Z− +m. +In other words, Z+ +m denotes the mth roots of unity, and Z− +m the mth roots of −1. +Theorem 3.13. For every n, there exists a tree domain Wn rooted at Vn, an integer m = +m(n), and a K-quasiconformal mapping ψ : Wn → D∗ so that: +(1) ψ is C-vertex-supported for some C > 0, and ψ(z) = z off of supp(ψz), +(2) on any edge e of ∂Wn ∩ T, ψ is length-multiplying and ψ(e) is an edge in T \ Zm, +(3) for any edge e of ∂Wn ∩D∗, ψ(e) consists of two edges in T\Zm. Moreover, if x ∈ e, +the two limits limWn∋z→x ψ(z) ∈ T are equidistant from Z+ +m, and from Z− +m, and +(4) C, K do not depend on n. +Proof. We consider the 2π-periodic covering map +(3.6) +φ := σ ◦ λ ◦ exp ◦(z �→ −iz) : H �→ Ω. + +Izl>1 +ov16 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +Figure 9. For any x ∈ ∂Wn ∩ D∗, there are two limits limWn∋z→x ψ(z) ∈ T +as illustrated in this figure. Theorem 3.13(3) says that these two limits are +equidistant from the nearest black vertex, and are equidistant from the nearest +white vertex. +inducing a periodic partition φ−1(Vn) of ∂H. +By (2.4), Definition 3.4, and Proposition +3.10(2), any two edges of H have comparable lengths with constant independent of n. There- +fore, Lemma 5.1 of [Bis15] applies to yield a 2π-periodic K-quasiconformal map Ψn of H +onto a subdomain Ψn(H) ⊊ H, with K independent of n. We let +Wn := exp(−iΨn(H)) +and +(3.7) +ψ := exp ◦ − iΨ−1 +n ◦ i log : Wn → D∗. +The map (3.7) is well-defined, and the conclusions of the theorem follow from Lemma 5.1 of +[Bis15]. +□ +Notation 3.14. We will use the notation Ω′ +n := (λ ◦ τ)−1(Wn). +4. Annular Interpolation Between the Identity and a Conformal Mapping +Recall from Notation 3.1 that we have fixed ε > 0, a compact set K, disjoint analytic +domains (Di)k +i=1 so that U := ∪iDi contains K, and f holomorphic in a neighborhood of U +with ||f||U < 1. In this section, we briefly define two useful interpolations in Lemmas 4.3 +and 4.4. +Since the domain Di contains the compact set K∩Di, the definitions and results of Section +2 apply to (ε, K ∩ Di, Di, f|Di) for each 1 ≤ i ≤ k (see Notation 2.7). Thus Remark 2.13 +applies to define (4.1), (4.2) and (4.3) in the following. +Definition 4.1. Let 1 ≤ i ≤ k. We define the Jordan curve +(4.1) +γi := γ(ε, K ∩ Di, Di, f|Di). +Recalling that int(γi) denotes the bounded component of �C \ γi, we define +(4.2) +Ψi := Ψ(ε, K ∩ Di, Di, f|Di) +to be a Riemann mapping Ψi : D → int(γi). Lastly, we define the finite Blaschke products +(4.3) +Bn := Bn(ε, K ∩ Di, Di, f|Di) on Di, + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +17 +Figure 10. Illustrated is the map ηΨ +i : D∗ → �C \ Ψi(riD) of Lemma 4.3 in +the case ri = 1. The dotted circle on the right depicts the unit circle. +where we suppress the dependence of (Bn)∞ +n=1 on i from the notation. +Recall that in Section 3, we defined curves {Γi}k−1 +i=1 connecting the domains Di, and in +Notation 3.5 we fixed a component Ω of the complement of U ∪ ∪k−1 +i=1 Γi. +Notation 4.2. After relabeling the (Di)k +i=1 if necessary, there exists 1 ≤ ℓ ≤ k so that +∂Di ∩ ∂Ω ̸= ∅ if and only if i ≤ ℓ (see Figure 6 for example). For each 1 ≤ i ≤ ℓ, note that +the intersection ∂Di ∩ ∂Ω consists of a single Jordan curve. We let ri := |Bn(∂Di ∩ ∂Ω)|, so +that 1 − ε ≤ ri ≤ 1 by (2.1). +The two interpolations we will need are given in Lemmas 4.3 and 4.4 below. In Lemma +4.3, we define an interpolation ηΨ +i between z �→ z on |z| = 2 with z �→ Ψi(riz) on |z| = 1 +(see Figure 10), and in Lemma 4.4 we modify ηΨ +i to define a map ηi so that ηi(z) = ηi(z) for +|z| = 1. +Lemma 4.3. For each 1 ≤ i ≤ ℓ, there is a quasiconformal mapping ηΨ +i : D∗ → �C \ Ψi(riD) +satisfying the relations: +(4.4) +ηΨ +i (z) = z for |z| ≥ 2 and +(4.5) +ηΨ +i (z) = Ψi(riz) for all |z| = 1. +Moreover, if Di, Dj for 1 ≤ i, j ≤ ℓ are connected by one of the curves (Γi)k−1 +i=1 , then +(4.6) +ηΨ +i ([−2, −1]) ∩ ηΨ +j ([1, 2]) = ηΨ +i ([1, 2]) ∩ ηΨ +j ([−2, −1]) = ∅. +Proof. The existence of ηΨ +i +satisfying (4.4) and (4.5) follows from a standard lemma on +the extension of quasisymmetric maps between boundaries of quasiannuli (see, for instance, +Proposition 2.30(b) of [BF14]). If (4.6) fails for the collection (ηΨ +i )i∈I thus defined, we can +renormalize the conformal mappings (Ψi)ℓ +i=1 appropriately (to rotate the points Ψi(±ri) +along the curve Ψi(riT)), and post-compose a subcollection of the ηΨ +i by diffeomorphisms of + +DV +n!18 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +2D \ Ψi(riD) so that (4.6) is satisfied for the composition when i, j ∈ I, and (4.4) and (4.5) +still hold. +□ +Lemma 4.4. For each 1 ≤ i ≤ ℓ, there is a quasiconformal mapping +ηi : D∗ → C \ Ψi([−ri, ri]) +satisfying the relations +(4.7) +ηi(z) = z for |z| ≥ 2, +(4.8) +ηi(z) = ηi(z) for |z| = 1, and +(4.9) +ηi(z) = ηΨ +i (z) for z ∈ R ∩ D∗. +Proof. Define +γ+ +i := ηΨ +i (∂(A(1, 2) ∩ H)). +Let η be a quasisymmetric mapping of T∩H onto [−1, 1] fixing ±1 (one can take η := M|T∩H +where M is a Mobius transformation mapping −1, 1, i to −1, 1, 0, respectively). Define a +mapping g on γ+ +i by: +(4.10) +g(z) := +� +Ψi ◦ η ◦ Ψ−1 +i (z) +z ∈ Ψi(T ∩ H) +z +otherwise +Since g is a quasisymmetric mapping, a standard lemma on extension of quasisymmetric +maps between boundaries of quasidisks (see, for instance, Proposition 2.30(a) of [BF14]) +implies that g may be extended to a quasiconformal mapping of ηΨ +i (A(1, 2) ∩ H). Define g +similarly in ηΨ +i (A(1, 2) ∩ (−H)). We let ηi := g ◦ ηΨ +i . It is then straightforward to check that +ηi satisfies (4.7)-(4.9). +□ +Remark 4.5. Lemmas 4.3 and 4.4 define 2ℓ many quasiconformal mappings: {ηΨ +i }ℓ +i=1 and +{ηi}ℓ +i=1. The definition of the mappings ηΨ +i , ηi depend on the objects ε, K, (Di)k +i=1, f as +fixed in Notation 3.1, but not on the parameter n in (4.3). Thus we record the trivial but +important observation that the mappings {ηΨ +i }ℓ +i=1 and {ηi}ℓ +i=1 are quasiconformal with a +constant independent of n. +5. Annular Interpolation Between a Blaschke Product and a Power Map +Recall that we have fixed ε > 0, a compact set K, disjoint analytic domains (Di)k +i=1 so +that U := ∪iDi contains K, and f holomorphic in a neighborhood of U with ||f||U < 1. The +curves {Γi}k−1 +i=1 connect the domains (Di)k +i=1, and Ω is a component of the complement of +U ∪ ∪k−1 +i=1 Γi with τ : Ω → D∗ conformal. Recall that the domain Ω′ +n was defined in Theorem +3.13 and Notation 3.14 by removing from Ω a collection of trees rooted at the vertices along +∂Ω, and the map ψ ◦ λ ◦ τ maps Ω′ +n onto D∗ (see Proposition 3.10 and Theorem 3.13). + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +19 +Notation 5.1. Recall from Notation 4.2 that ∂Di ∩ ∂Ω ̸= ∅ if and only if 1 ≤ i ≤ ℓ. Hence +exactly ℓ − 1 of the curves (Γi)k−1 +i=1 intersect ∂Ω. By relabelling the (Γi)k−1 +i=1 if necessary, we +may assume Γj intersects ∂Ω if and only if 1 ≤ j ≤ ℓ − 1. +Let m = m(n) be as in Theorem 3.13. To prove our main results, we will need to modify +z �→ zm in D∗ so that, roughly speaking, (z �→ zm) ◦ ψ ◦ λ ◦ τ(z) agrees with the Blaschke +products Bn (see Definition 4.1) along ∂Di. This is done in Theorem 5.3 below. Its proof +uses the following. +Proposition 5.2. Suppose φ1, φ2 are C1 homeomorphisms of a C1 Jordan arc e such that: +(1) φ1(e) = φ2(e), +(2) φ1, φ2 agree on the two endpoints of e, and +(3) |φ′ +1(z)| = |φ′ +2(z)| for all z ∈ e. +Then φ1 = φ2 on e. +The proof of Proposition 5.2 is a consequence of the Fundamental Theorem of Calculus and +is left to the reader. Recall the constant ri := |Bn(∂Di ∩ ∂Ω)| of Notation 4.2. +Theorem 5.3. For every n, there exists a locally univalent K-quasiregular mapping hn : +D∗ → D∗ so that: +(1) hn(z) = zm for |z| ≥ +m√ +2 where m := m(n) is as in Theorem 3.13, +(2) hn ◦ ψ ◦ λ ◦ τ(z) = Bn(z)/ri for every z ∈ ∂Di and 1 ≤ i ≤ l, and +(3) K is independent of n. +Proof. Fix the standard branch of log. Given an edge e ∈ ∂Di, we have by Theorem 3.13 +that +(5.1) +log ◦ψ ◦ λ ◦ τ(e) = {0} × +�jπ +m , (j + 1)π +m +� +for some 0 ≤ j ≤ 2m − 1. +Denote the vertical line segment in (5.1) by ve. Let f : ve �→ e be a length-multiplying, C1 +homeomorphism so that f −1 agrees with log ◦ψ ◦ λ ◦ τ on the two endpoints of e. Consider +the maps: +(5.2) +z �→ mz for z ∈ +�log 2 +m +� +× +�jπ +m , (j + 1)π +m +� +, +(5.3) +z �→ log ◦r−1 +i Bn ◦ f for z ∈ {0} × +�jπ +m , (j + 1)π +m +� +. +For each 1 ≤ i ≤ l, the Blaschke products Bn are orientation-preserving on the unique outer +boundary component of Di, and orientation-reserving on all other boundary components of +Di. This implies that we may choose the branch of log in (5.3) so that the images of (5.2) +and (5.3) are horizontal translates of one another (recall Bn(e) is a circular arc of angle +π), and the derivative of (5.3) is strictly positive for all z ∈ ve. Since the derivative of + +20 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +Figure 11. Illustrated is the proof of Theorem 5.3. In logarithmic coordi- +nates the desired interpolation is denoted φ, and hn is then defined by (5.5) +and (5.6). +(5.2) is also strictly positive, this means the linear interpolation between (5.2) and (5.3) is a +homeomorphism. +By (2.5), we have that |B′ +n| is comparable at all points of e with constant independent of e +and n. Thus, since f is length-multiplying and log is length-multiplying on Euclidean circles +centered at 0, we conclude that the derivative of (5.3) is comparable to m at all points of +ve with constant independent of e and n. Thus, we conclude that the linear interpolation +between (5.2) and (5.3) in the rectangle +(5.4) +� +0, log 2 +m +� +× +�jπ +m , (j + 1)π +m +� +is K-quasiconformal with K independent of e and n (see, for instance, Theorem A.1 of +[MPS20]). Denote the linear interpolation by φ (see Figure 11). +We define +(5.5) +hn := exp ◦φ ◦ log in {z ∈ D∗ : z/|z| ∈ ψ ◦ λ ◦ τ(e)} ∩ {|z| ≤ +m√ +2}. +The equation (5.5) defines hn(z) for z in {z : 1 ≤ |z| ≤ +m√ +2} and sharing a common angle +with the image under ψ ◦ λ ◦ τ of an edge on some ∂Di. We finish the definition of hn by +simply setting: +(5.6) +hn(z) := zm in {z ∈ D∗ : z/|z| ∈ ψ ◦ λ ◦ τ(∂Ω′ +n \ (∪i∂Di))}. +The conclusion (1) now follows by definition of hn, and (3) follows since hn is a composition +of holomorphic mappings and a K-quasiconformal interpolation where we have already noted +that K is independent of n. + +h +nA GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +21 +We now show that conclusion (2) follows from Proposition 5.2. Fix an edge e on ∂Di. +Recall ve := log ◦ψ ◦ λ ◦ τ(e). Thus, by (5.3) and (5.5) we have that: +(5.7) +hn ◦ ψ ◦ λ ◦ τ = r−1 +i Bn ◦ f ◦ log ◦ψ ◦ λ ◦ τ on e. +First note that (5.7) agrees set-wise with r−1 +i Bn on e and at the endpoints of e. The map +ψ ◦ λ ◦ τ is length-multiplying (by Proposition 3.10(3) and Theorem 3.13(2)), log is length- +multiplying on the circular segment ψ ◦ λ ◦ τ(e), and f is length-multiplying by definition. +Thus the modulus of the derivative of f ◦log ◦ψ◦λ◦τ is constant on e, and so the derivatives +of (5.7) and r−1 +i Bn have the same modulus at each point of e. Conclusion (2) now follows +from Proposition 5.2. +□ +6. Joining Different Types of Boundary Arcs: the Map En +Recall that in Section 4 we defined the maps ηΨ +i , ηi where 1 ≤ i ≤ ℓ, and in Section 5 +we defined the map hn for all n ∈ N. In this section we define a map En in D∗ which is +roughly given by either z �→ ηΨ +i ◦ hn(z) or z �→ ηi(z) ◦ hn(z), where i is allowed to depend +on arg(z) and which of ηΨ +i , ηi we post-compose hn with is also allowed to depend on arg(z). +Thus, we will need a way to interpolate between the definitions of η�� +i , ηi, for different i. The +interpolation regions are defined in Definition 6.1 below, and the map En in Proposition 6.2. +It will be useful to keep Figure 12 in mind for the remainder of this section. +Definition 6.1. Mark one edge ei on Γi for each 1 ≤ i ≤ ℓ − 1. Label the ℓ components of +∂Ω′ +n \ ∪iei as (Gi)ℓ +i=1, where ∂Di ⊂ Gi. Let +(1) J D +i +denote those edges in ∂Di, +(2) J G +i +denote those edges in Gi \ J D +i , +(3) J e denote the edges (ei)ℓ−1 +i=1. +In other words, J D +i +are the edges shared by ∂Ω′ +n and ∂Di, J e consists of ℓ − 1 edges: one +on each of the curves (Γi)ℓ +i=1, and J G +i +are the remaining edges on Gi. Thus we have: +∂Ω′ +n = J e ∪ +� +i +� +J D +i +∪ J G +i +� +. +For z ∈ D∗, we define: +(6.1) +En(z) := +� +ηψ +i ◦ hn(z) +if +z/|z| ∈ ψ ◦ λ ◦ τ(J D +i ) +ηi ◦ hn(z) +if +z/|z| ∈ ψ ◦ λ ◦ τ(J G +i ) +It remains to define En(z) for z ∈ D∗ satisfying z/|z| ∈ ψ ◦ λ ◦ τ(J e). We do so in the +following Proposition. +Proposition 6.2. The map En extends to a locally univalent K-quasiregular mapping En : +D∗ → C satisfying En(z) = zm for |z| ≥ +m√ +2, where m = m(n) is as in Theorem 3.13. +Moreover, K does not depend on n. + +22 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +Figure 12. Illustrated is Definition 6.1. The curves Γ1, Γ2 are depicted as +black dotted lines, except for the edges e1 ⊂ Γ1, e2 ⊂ Γ2 which are in thick +black. +Proof. Consider (6.1). Note that if En is defined at z and |z| ≥ +m√ +2, then En(z) = zm +by Theorem 5.3(1) and (4.4), (4.7). Thus, setting En(z) := zm for |z| ≥ +m√ +2 extends the +definition of En. +It remains to extend the definition of En to: +(6.2) +{z : 1 ≤ |z| ≤ +m√ +2 and z/|z| ∈ ψ ◦ λ ◦ τ(ei)}, for 1 ≤ i ≤ ℓ − 1. +Each of the ℓ − 1 sets in (6.2) consists of 2 quadrilaterals which we denote by Q± +i . The +curve Γi connects two distinct elements of (Di)ℓ−1 +i=1. In order to avoid complicating notation +significantly, we will assume without loss of generality that Γi connects Di to Di+1. Let +γi ⊂ 2D be a smooth Jordan arc connecting ηΨ +i (1) to ηΨ +i+1(−1). Moreover, by (4.6), we can +choose γi so that the union of the arcs +(6.3) +ηΨ +i ([1, 2]), 2T ∩ H, ηΨ +i+1([−2, −1]), γi +forms a topological quadrilateral we denote by Q+ +i (in particular none of the arcs in (6.3) +intersect except at common endpoints). +Define a quasisymmetric homeomorphism g+ +i : ∂Q+ +i → ∂(A(1, 2) ∩ H) (see Figure 13) by +g+ +i (z) = z for z ∈ 2T +g+ +i (z) = (ηΨ +i )−1(z) for z ∈ ηΨ +i ([1, 2]) +g+ +i (z) = (ηΨ +i+1)−1(z) for z ∈ ηΨ +i+1([−2, −1]), +and extending g+ +i to a quasisymmetric homeomorphism of γi to T ∩ H. The mapping g+ +i +extends to a quasiconformal homeomorphism g+ +i : Q+ +i → A(1, 2) ∩ H (see Lemma 2.24 of + +D1 +" +G1A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +23 +Figure 13. Illustrated is the quadrilateral Q+ +i and the map g+ +i in the proof +of Proposition 6.2. +[BF14]). We define En(z) := (g+ +i )−1(zm) for z ∈ Q+ +i . A similar definition of g− +i : Q− +i → +A(1, 2) ∩ −H is given (using the same curve γi) so that +g+ +i (z) = g− +i (z) for z ∈ T ∩ H. +We let En(z) := (g− +i )−1(zm) for z ∈ Q− +i . +To summarize, we have defined En in each of the three regions +{z ∈ D∗ : z/|z| ∈ ψ ◦ λ ◦ τ(J D +i )}, +(6.4) +{z ∈ D∗ : z/|z| ∈ ψ ◦ λ ◦ τ(J G +i )}, +(6.5) +{z ∈ D∗ : z/|z| ∈ ψ ◦ λ ◦ τ(J e +i )}, +(6.6) +Indeed, the definition of En in (6.4) and (6.5) was given already in (6.1), and in this proof we +have defined En in (6.6). The definitions of En in each of (6.4), (6.5), (6.6) agree along any +common boundary, and thus by removability of analytic arcs for quasiregular mappings, it +follows that En is quasiregular on D∗. Moreover, En has no branched points in D∗, and hence +En is locally quasiconformal. The dilatation of the map En depends only on the dilatation +of hn (which is independent of n by Theorem 5.3(3)) and the dilatations of the the finite +collection of quasiconformal maps used in its definition: ηΨ +i , ηi, g+ +i , g− +i , and hence we may +take K independent of n. +□ +7. Defining gn in Ω′ +n +First we recall our setup. +We have fixed ε > 0, a compact set K, disjoint, analytic +domains (Di)k +i=1 so that K ⊂ U := ∪iDi, and f holomorphic in a neighborhood of U with +||f||U < 1. We defined curves {Γi}k−1 +i=1 connecting the domains (Di)k +i=1, and we denoted by +Ω a component of the complement of U ∪ ∪k−1 +i=1 Γi with τ : Ω → D∗ conformal. The domain +Ω′ +n is contained in Ω, and ψ ◦ λ ◦ τ maps Ω′ +n onto D∗. In Section 6 we defined the map En. +Definition 7.1. We define the mapping gn : Ω′ +n → �C by +(7.1) +gn := En ◦ ψ ◦ λ ◦ τ. + +Izl=2 +Izl=2 +Izl=1 +Di +D +i+124 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +We will now record at which points the function gn|Ω′n is locally n : 1 for n > 1. +Definition 7.2. Let g be a quasiregular function, defined in a neighborhood of a point +z ∈ C. We say that z is a branched point of g if for any sufficiently small neighborhood U +of z, the map g|U is n : 1 onto its image for n > 1. We say w ∈ C is a branched value of g +if w = g(z) for a branched point z of g. We denote the branched points of a quasiregular +mapping g by BP(g), and the branched values by BV(g). +Remark 7.3. Recall that in Notation 3.5 we fixed a point p ∈ Ω satisfying τ(p) = ∞. +Proposition 7.4. The mapping gn : Ω′ +n → C of Definition 7.1 is K-quasiregular and C- +vertex supported for K, C independent of n. Moreover, g−1 +n (∞) = {p}, +(7.2) +BP(gn) ⊂ +� +e∈∂Ωn +{z : dist(z, e) < C · diam(e)}, and +(7.3) +BV(gn) ⊂ +k� +i=1 +Ψi(riT). +Proof. Since each of the mappings in the composition (7.1) are K-quasiregular and C-vertex +supported for K, C independent of n, the same is true of gn. The only points where the +mapping gn is locally l : 1 for l > 1 are a subset of the vertices of the graph ∂Ω′ +n. By +Theorem 3.13, the vertices of ∂Ω′ +n all lie in +� +e∈∂Ωn +{z : dist(z, e) < C · diam(e)}. +Thus, (7.2) is proven. Moreover, any vertex of ∂Ω′ +n is mapped to a point on one of the curves +Ψi(riT) by gn. Hence, (7.3) follows since BV(gn) = gn(BP(gn)). It remains to show: +(7.4) +g−1 +n (∞) = {p}. +Indeed, note that En◦ψ◦λ fixes ∞ and has no finite poles. The map τ : Ω → D∗ is conformal +and hence only one point p is mapped to ∞. The relation (7.4) now follows. +□ +It will be useful to record the following result. +Proposition 7.5. Let r > 1. Then for all sufficiently large n, we have: +(7.5) +gn(z) = τ(z)m for any z ∈ τ −1({z : |z| > r}). +Proof. Consider the functional equation (7.1) defining gn. +The maps λ, ψ are vertex- +supported, and moreover λ (respectively, ψ) is the identity outside of the support of λz, +(respectively, ψz). By Proposition 3.7, we therefore have that ψ ◦ λ(z) = z if z ∈ τ −1({z : +|z| > r}) and n is sufficiently large. The relation (7.5) now follows from (7.1) and Theorem +5.3(1) since m → ∞ as n → ∞. +□ + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +25 +Remark 7.6. As in Remark 2.13, we note that our Definition 7.1 of gn is determined by a +choice of the objects K, U, D, f, ε, Ω, p we fixed in Notations 3.1 and 3.5. When we wish +to emphasize this dependence, we will write gn(K, U, D, f, ε, Ω, p). In particular, it will be +useful in the next section to think of gn as a function taking as input any choice of K, U, +D, f, ε, Ω, p satisfying the conditions in Notations 3.1, 3.5, and outputting (via Definition +7.1) a quasiregular function gn(K, U, D, f, ε, Ω, p) defined on Ω′ +n. +8. Verifying gn is Quasiregular on �C +In this section we combine our efforts in Sections 2-7 to define an approximant gn : �C → �C +of a given f. The approximant gn will not be holomorphic as required in Theorems A and +B, but we will solve this problem in the next section by applying the Measurable Riemann +Mapping Theorem. We fix the following for Sections 8-9. +Notation 8.1. Fix K, f, D, ε, P as in the statement of Theorem B. Denote by U the +neighborhood of K in which f is holomorphic. Define +(8.1) +P ′ := {p ∈ P : p is contained in a component V of �C \ K such that V ̸⊆ U}. +Compactness of K implies that U contains all but finitely many components of �C \ K, and +so the set P ′ is finite. Moreover, P ′ does not depend on ε. By shrinking U if necessary, we +may assume that: +(1) U ∩ P ′ = ∅, +(2) P ′ contains exactly one point in each component of �C \ U, +(3) f is holomorphic in a neighborhood of U ⊂ D, and +(4) the components of U are a finite collection of analytic Jordan domains (Di)k +i=1 so +that (2.6) holds for each Di. +Let K′ be a compact set such that K ⊂ int(K′) ⊂ K′ ⊂ U. We will assume for now that +||f||U < 1. +We now define a quasiregular approximation gn of f by applying the Blaschke product +construction of Section 2 in each Di, and by applying the folding construction of Sections +3-7 in each complementary component of ∪i(Di ∪ Γi): +Definition 8.2. For every n, we define a quasiregular mapping gn as follows. Recalling +Remark 2.13, we first set +(8.2) +gn := gn(ε, K′ ∩ Di, Di, f|Di) in Di for 1 ≤ i ≤ k +The equation (8.2) defines the curves (Γi)k +i=1 by way of Proposition 3.3, and we enumerate +the components of +�C \ +� +U ∪ +k−1 +� +i=1 +Γi +� + +26 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +by (Ω(i))ℓ +i=1. Recalling Remark 7.6 and Notation 3.14, we extend the definition of gn to the +open set +(8.3) +Ω := �C \ +� ℓ� +i=1 +∂Ω′ +n(i) +� +by the formula +(8.4) +gn := gn(K′, U, D, f, ε, Ω(i), P ′ ∩ Ω(i)) in Ω′ +n(i) for 1 ≤ i ≤ ℓ. +Proposition 8.3. The quasiregular function gn is C-vertex supported and K-quasiregular +for C, K independent of n. +Proof. For gn|Ω′n(i) this is exactly Proposition 7.4, and so the conclusion follows since gn is +holomorphic in U. +□ +The function gn is now defined on all of �C except for the edges of each ∂Ω′ +n(i). We show in +Propositions 8.4, 8.5 below that gn in fact extends continuously across each edge of ∂Ω′ +n(i), +and deduce in Corollary 8.6 that gn extends quasiregularly across ∂Ω. +Proposition 8.4. The K-quasiregular function gn : Ω → �C extends to a continuous function +gn : Ω ∪ e → �C for any edge e ⊂ ∂Ω ∩ ∂U. +Proof. Let i be so that e ⊂ ∂Di and denote the unique element of (Ω(i))ℓ +i=1 that contains e +on its boundary by Ω(j). Recall by Definitions 2.12 and 7.1 that +(8.5) +gn|Di = Ψi ◦ Bn, +(8.6) +gn|Ω′n(j) = En ◦ ψ ◦ λ ◦ τ. +Assume that i ∈ I (the reasoning in the case i ̸∈ I will be the same), so that by Theorem +5.3(2) we have +hn ◦ ψ ◦ λ ◦ τ = r−1 +i Bn on e. +By (4.5) and the definition (6.1) of En, it follows from (8.6) that +gn|Ω′n(j)(z) = Ψi ◦ Bn(z) for z ∈ e, +in other words gn|Ω′n(j) and gn|Di agree pointwise on e. +□ +Proposition 8.5. The K-quasiregular function gn : Ω → �C extends to a continuous function +gn : Ω ∪ e → �C for any edge e ⊂ ∂Ω ∩ (∪ℓ +i=1Ω(i)). +Proof. Let j be so that e ⊂ ∂Ω′ +n(j), and as in the proof of Proposition 8.4, recall that +(8.7) +gn|Ω′n(j) = En ◦ ψ ◦ λ ◦ τ. +Let x ∈ e. There are two limits +lim +Ω′n(j)∋z→x ψ ◦ λ ◦ τ(z), + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +27 +each lying on the unit circle. Denote them by ζ±. By Theorem 3.13(3), +ζm ++ = ζm +− . +Thus, by (4.8) and (6.1), we conclude that there is a unique limit +lim +Ω′n(j)∋z→x En ◦ ψ ◦ λ ◦ τ(z). +Hence, setting +gn(x) := +lim +Ω′n(j)∋z→x En ◦ ψ ◦ λ ◦ τ(z) +defines a continuous extension of gn across the edge e. +□ +Corollary 8.6. The K-quasiregular function gn : Ω → �C extends to a K-quasiregular func- +tion gn : �C → �C. +Proof. The set �C \ Ω = ∂Ω consists of a finite collection of analytic arcs: the edges of +the graphs ∂Ω′ +n(i) over 1 ≤ i ≤ ℓ. Thus, by removability of analytic arcs for quasiregular +mappings, it suffices to show that gn : Ω → �C extends continuously across each such edge. +There are two types of edges to check: those that lie on the boundary of a domain Di, and +those that lie in the interior of a domain Ω(i). We have already checked continuity across +both types of edges in Propositions 8.4, 8.5, and so the proof is complete. +□ +9. Proof of the Main Theorems +In Section 9 we prove Theorems A and B, modulo the proof of Theorem 2.6 which is left +to Sections 10-12. +Recall that in Section 8 we fixed the objects K, f, D, ε, P as in Theorem B (see Notation +8.1), and we defined a quasiregular approximation gn to f in Definition 8.2. We also showed +in Section 8 that gn in fact extends to a quasiregular function gn : �C → �C. Now we apply +the MRMT below in Definition 9.1 to obtain the rational maps rn : �C → �C which we will +prove satisfy the conclusions of Theorems A and B for large n. +Definition 9.1. The mapping gn induces a Beltrami coefficient µn := (gn)z/(gn)z, which, by +way of the MRMT, defines a quasiconformal mapping φn : �C → �C such that rn := gn ◦ φ−1 +n +is holomorphic. We normalize φn so that φn(∞) = ∞ and φn(z) = z + O(1/|z|) as z → ∞. +We now begin deducing that for large n, the maps rn satisfy the various conclusions in +Theorems A and B. +Proposition 9.2. The function rn of Definition 9.1 is rational, and r−1 +n (∞) = φn(P ′). In +particular, if K is full and P = {∞}, then rn is a polynomial. +Proof. The function rn is holomorphic on �C and takes values in �C: the only such functions are +rational. Note that g−1 +n (∞)∩U = ∅ since gn is bounded on U. Thus, by Proposition 7.4 and +(8.4), we have that g−1 +n (∞) = P ′. Since rn := gn ◦ φ−1 +n , we conclude that r−1 +n (∞) = φn(P ′). + +28 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +The last statement of the proposition follows since we normalized φn(∞) = ∞, and the only +rational functions with a unique pole at ∞ are polynomials. +□ +Proposition 9.3. For all R < ∞, the mapping φn satisfies: +(9.1) +||φn(z) − z||R·D +n→∞ +−−−→ 0. +Proof. Since gn is C-vertex supported by Proposition 8.3, we conclude from Proposition 3.7 +that +(9.2) +Area(supp(µn)) +n→∞ +−−−→ 0. +The relation (9.1) now follows from (9.2) since ||µn||L∞ ≤ K for all n by Proposition 8.3. +□ +Theorem 9.4. For all sufficiently large n, the mapping rn satisfies CP(rn) ⊂ D. +Proof. By Proposition 3.7, we have +max +� +diam(e) : e is an edge of +ℓ� +i=1 +∂Ω(i) +� +n→∞ +−−−→ 0. +Thus, since D is a domain containing ∪ℓ +i=1∂Ω(i), we have by (7.2) that BP(gn) \ U ⊂ D for +large n. Since U ⊂ D we conclude that BP(gn) ⊂ D for large n. The result now follows from +Proposition 9.3 since φ(BP(gn)) = CP(rn). +□ +Theorem 9.5. For all sufficiently large n, we have +||rn − f||K < 3ε. +Proof. First we note that since f is uniformly continuous on K′, there exists δ > 0 so that +if z, w ∈ K′ and |z − w| < δ, then |f(z) − f(w)| < ε. By Proposition 9.3, we can conclude +that +(9.3) +||φn(z) − z||K < min(δ, dist(K, ∂K′)) +for all sufficiently large n. +Let z ∈ K, w := φ−1 +n (z) and suppose j is such that z ∈ Dj. Then +|rn(z) − gn(z)| = |gn(w) − gn(z)| ≤ |gn(w) − f(w)| + |f(w) − f(z)| + |f(z) − gn(z)|. +(9.4) +Let +C := sup +z∈D +1≤i≤k +|Ψ′ +i(z)|. +Then +||gn − f||K′ ≤ C · ||Ψ−1 +j +◦ gn − Ψ−1 +j +◦ f||K′, +and we deduce by (2.15) that: +||gn − f||K′ ≤ C · ||Bn − Ψ−1 +j +◦ f||K′. + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +29 +Applying Theorem 2.10 (see also Remark 2.9), we conclude that +(9.5) +||gn − f||K′ +n→∞ +−−−→ 0. +Next, we deduce from (9.3), (9.4) and (9.5) that +|rn(z) − gn(z)| < 2ε +for sufficiently large n. It follows that for sufficiently large n: +||rn − f||K ≤ ||rn − gn||K + ||gn − f||K < 3ε. +□ +Theorem 9.6. For all sufficiently large n we have CV(f|K) ⊂ CV(rn). +Proof. Let z ∈ CP(f|K), and let i be so that z ∈ Di. Then Ψ−1 +i ◦f(z) ∈ CV(Bn) by Theorem +2.10. Thus f(z) ∈ CV(Ψi ◦ Bn). Thus, by the Definition 2.12 of gn, we have for large n that +f(z) ∈ BV(gn) = CV(rn). +□ +Theorem 9.7. For all sufficiently large n, we have +CV(rn) ⊂ fill{z : d(z, f(K)) < ε}. +Proof. Since +CV(rn) = BV(gn), +it suffices to show that for every z ∈ BP(gn) and sufficiently large n, we have +(9.6) +gn(z) ∈ fill{z : d(z, f(K)) < ε}. +For z ∈ BP(gn) \ U, (9.6) follows from (7.3). +For z ∈ BP(gn) ∩ Di, (9.6) follows from +Definition 2.12 of gn|Di. +□ +Proof of Theorem B: In the special case that ||f||K < 1, we have already proven that the +mappings rn satisfy the conclusions of Theorem B for all sufficiently large n. Indeed, Theorem +9.5 says that ||rn − f||K < ε, conclusion (2) in Theorem B is Theorem 9.4, and conclusion +(3) is Theorems 9.6, 9.7. Conclusion (1) follows from Propositions 9.2, 9.3. The general case +follows by applying the above special case to an appropriately rescaled f. +□ +Proof of Theorem A: When K is full, we may take P = {∞} and apply Theorem B, in which +case Proposition 9.2 guarantees that the maps rn are polynomials. +□ +Theorem 9.8. (Mergelyan+) Let K ⊂ C be full, suppose f ∈ C(K) is holomorphic in +int(K), and let D be a domain containing K. For every ε > 0, there exists a polynomial p +so that ||p − f||K < ε and: +(1) CP(p) ⊂ D, +(2) CV(p) ⊂ fill{z : d(z, f(K)) ≤ ε}. + +30 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +Proof : By the usual version of Mergelyan’s Theorem, there exists a polynomial q so that +||q − f||K < ε/2. Apply Theorem A to K, D, q, ε/2 to obtain an approximant of q which +we denote by p. The polynomial p satisfies the conclusions of Theorem 9.8. +□ +Corollary 9.9. (Weierstrass+) Suppose that I ⊂ R is a closed interval, f : I → R is +continuous, and U, V ⊂ C are planar domains containing I, f(I), respectively. Then, for +every ε > 0, there exists a polynomial p with real coefficients so that ∥f − p∥I ≤ ε, and +(1) CP(p) ⊂ U, +(2) CV(p) ⊂ V . +Proof. Let I = [a, b], and f, U, V as in the statement of the corollary. By Theorem 9.8, +there exists a complex polynomial q so that ||q − f||[a,b] < ε/2. The real polynomial +Q(z) := q(z) + q(z) +2 +satisfies Q(x) = Re(q(x)) for x ∈ R and hence ||Q − f||[a,b] < ε/2. We will use the symbol +⋐ to mean compactly contained. Let V1 ⋐ V be a sufficiently small, R-symmetric domain +containing f(I) so that there is a component of Q−1(V1) (which we denote by U1) satisfying +U1 ⋐ U. Let U2 be a R-symmetric, analytic domain satisfying I ⋐ U1 ⋐ U2 ⋐ U. Recall +Notation 8.1 and consider: +(1) the compact set U1, +(2) the analytic function Q, +(3) the analytic domain U2 containing U1, +(4) min{ε/2, dist(∂V1, ∂V )}, +(5) P = {∞}. +Applying Definition 8.2 to (1)-(5) yields quasiregular mappings gn with R-symmetric Bel- +trami coefficient, so that +(9.7) +pn := gn ◦ φ−1 +n +is a real polynomial approximant of Q for large n satisfying: +(1) ||pn − f||[a,b] < ε, +(2) CP(pn) ⊂ U2, +(3) CV(pn) ⊂ V . +Thus pn satisfies the conclusion of Corollary 9.9 for large n. +□ +Remark 9.10. If we make further assumptions on the compact set K, the conclusion +CV(r) ⊂ fill{z : d(z, f(K)) < ε} +of Theorems A and B can be improved to +(9.8) +CV(r) ⊂ fill(f(K)), + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +31 +which is equivalent to +CV(r) ⊂ f(K) +if f(K) is full. Indeed, if for instance the interiors of K, f(K) are analytic domains and +f : int(K) → int(f(K)) is proper, then a similar strategy as in the proofs of Theorems A +and B but replacing Ψ in (2.15) with a conformal map D �→ int(K) can be used to prove +(9.8). +Recall the notation Ω(i) from Definition 8.2, and let τi : Ω(i) → D∗ be the conformal +mapping satisfying τ −1 +i +(∞) = P ′ ∩ Ω(i) as in Notation 3.5. The following fact justifies part +of our description in the introduction of the behavior of the rational approximants off K. +Proposition 9.11. Let 1 < r < R < ∞. Then, for all sufficiently large n, we have +(9.9) +rn ◦ φn(z) = τi(z)m and +(9.10) +|rn(z)| > R +for all z ∈ τ −1 +i +({z : |z| > r}). +Proof. Fix R < ∞ and r > 1. From (7.5) and the functional equation (8.4) defining gn in +Ω(i), it follows that: +(9.11) +gn(z) = τi(z)m for all z ∈ τ −1 +i +({z : |z| > (r + 1)/2}) +for all large n. Since +(9.12) +rn ◦ φn = gn, +The relation (9.9) follows. Moreover, we have by Proposition 9.3 that: +(9.13) +φn ◦ τ −1 +i +({z : |z| > r}) ⊂ τ −1 +i +({z : |z| > (r + 1)/2}) +for all sufficiently large n. +Since ((r + 1)/2)m > R for large n, the relation (9.10) also +follows. +□ +10. Some Estimates Involving Harmonic Measure and Green’s Functions +In Sections 10-12 we turn to the proof of Theorem 2.6. In fact, we will prove a slightly +stronger result (Theorem 12.2 in Section 12) from which Theorem 2.6 follows. We begin by +recalling a few standard facts, and sketch the proofs for the convenience of the reader. Let +D(z, r) denote the disk of radius r centered at z. Lemma 10.1 is illustrated in Figure 14. +Lemma 10.1. If K ⊂ D is continuum connecting 0 to T, then ω(z, K, D \ K) ≥ c > 0 for +all |z| < 1/4 and some c > 0 independent of K. + +32 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +Figure 14. Illustrated is the compact set K of Lemma 10.1. +Proof. Exercise III.10 of [GM08] says that if E is a continuum connecting {|z| = 1 +2} to T +in D, then ω(0, E, D \ E) ≥ c = +2 +π tan−1(1/ +√ +8). +Apply the exercise and the maximum +principle to the disk D = D(z, 1 +2) to deduce the lemma. [GM08] gives a simple direct proof +of the exercise, but it also follows from Beurling’s projection theorem, e.g., Theorem II.9.2 +of [GM08]. +□ +Lemma 10.2. Suppose u, v are harmonic functions on D such that supD |u|, supD |v| ≤ M +and that |u − v| < ε on some continuum K connecting 0 to T. Then |u − v| ≤ εcM 1−c on +D(0, 1/4), where c > 0 is the constant from Lemma 10.1. +Proof. Consider the subharmonic function log |u−v|. It is less than log ε on K and less than +log M on ∂D. Thus for |z| ≤ 1/4, Lemma 10.1 implies +log |u(z) − v(z)| +≤ +ω(z, K, D \ K) log ε + ω(z, T, D \ K) log M +≤ +c log ε + (1 − c) log M, +so |u(z) − v(z)| ≤ εcM 1−c, as desired. +□ +Lemma 10.3. Suppose Ω is a planar domain, K ⊂ Ω is compact and connected, and 0 < +ε, M < ∞. Then there is a δ > 0 so that if h = u + i�u is holomorphic on Ω, �u vanishes at +some point of K, supΩ |h| ≤ M and supK |u| < δ, then supK |�u| < ε. +Proof. If K is single point, this is trivial since �u = 0 there by assumption, so assume K is +non-trivial. Choose η > 0 so that η < dist(K, ∂Ω) and η < diameter(K). Then for any +radius η disk D centered at a point of K, u is less than δ on a continuum connecting the +center of D to its boundary. This implies |u| ≤ δcM 1−c (for c as in Lemma 10.1) on a +η/4-neighborhood of K. Thus |∇u| = O(δcM 1−c/η) on the (η/8)-neighborhood U of K (this +uses the Cauchy estimate for |∇u|, e.g., Theorem 2.4 of [ABR01]). Since |∇�u| = |∇u| and + +Iwl=1 +K +.0 +Z +wl=1/4A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +33 +Figure 15. Illustrated in gray is the set Uδ of Lemma 10.4. +U is path connected, this implies �u is within ε of zero on K if δ is small enough (depending +on ε, η, M and the diameter of U in the path metric). +□ +We will use this later in the situation that if u and v are harmonic functions on Ω that +are close enough on K ⊂ Ω, then f = exp(u + i�u) and g = exp(v + i�v) are holomorphic +functions on Ω that are close on K (if �v is chosen to agree with �u at some point of K). +Next, we recall the well known boundary Harnack inequality (e.g., see Theorem 7.18 of +[Mar19] or Exercise I.6 of [GM08]). +Lemma 10.4. Suppose u and v are positive harmonic functions on D which extend contin- +uously to the boundary T, and suppose furthermore that u and v are both equal to zero on an +arc I ⊂ T. For δ > 0 let Uδ = {z ∈ D : dist(z, T \ I) > δ} (see Figure 15). Then for z ∈ Uδ, +δ2 +4 · u(0) +v(0) ≤ u(z) +v(z) ≤ 4 +δ2 · u(0) +v(0). +Note that, under these conditions, u and v have well defined inward normal derivatives on +I. By letting z → T radially, the inequalities above imply that ∂u +∂n and ∂v +∂n are comparable +(with the same constants as above) at any point of Uδ ∩ T. +Corollary 10.5. Suppose u is a positive harmonic function on D which extends continuously +to the boundary T and that equals zero on an arc I ⊂ T. Suppose a, b ∈ I both have distance +> δ from T \ I. Then +δ2 +4 · ∂u +∂n(a) ≤ ∂u +∂n(b) ≤ 4 +δ2 · ∂u +∂n(a). +Proof. We simply compare u to a rotation of itself. Let v(z) = u( b +az). Then v and u both +vanish on J = I ∩ a +b · I, and a ∈ J is distance > δ from either endpoint of J. Hence the +normal derivatives of u and v at a are comparable by the boundary Harnack principle, and +hence so are the normal derivatives of u at a and b. +□ + +Us +134 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +Suppose Ω is an analytic domain (see Definition 2.5). For w ∈ Ω, let G(z, w) be the +Green’s function on Ω with pole at w, i.e., G is harmonic on Ω \ {w}, vanishes identically +on ∂Ω and G(z, w) + log |z − w| is bounded in a neighborhood of w. Our assumptions on Ω +imply that Ω is regular for the Dirichlet problem, and hence that the Green’s function exists +and is unique for any w ∈ Ω (e.g., see Sections II.1 and II.2 of [GM08]). +Lemma 10.6. Suppose Ω is an analytic domain. If r > 0 is small enough (depending only +on Ω), x ∈ ∂Ω, and y ∈ Ω \ D(x, r), then the normal derivative of the Green’s function with +pole at y has comparable size at all points of ∂Ω ∩ D(x, r/2) with a constant independent of +y and Ω. +Proof. Choose r small enough that W = D(x, r) ∩ Ω is a Jordan domain whose boundary +consists of a sub-arc γ of ∂Ω and an arc of the circle ∂D(x, r). Let �γ = ∂Ω ∩ D(x, r/2). +Choose a point z ∈ W that is about distance r from ∂W and choose a conformal map +ϕ : W → D taking z to 0. If r is small enough, ϕ extends analytically across γ to all of +D(x, r) and by the Koebe distortion theorem it has comparable derivative at all points of +�γ. Also, since γ and each component of γ \ �γ has harmonic measure with respect to z that +is bounded away from zero, the image arcs on T all have lengths bounded uniformly from +below. Thus u(z) = G(ϕ−1(z), y) is a positive harmonic function on D vanishing on ϕ(γ). +By Corollary 10.5 the normal derivatives of u are comparable at all points of ϕ(�γ). Since +the values of |ϕ′| are comparable at all points of �γ, we can deduce the lemma from the chain +rule. +□ +Corollary 10.7. Suppose Ω is an analytic domain. If r > 0 is small enough (depending only +on Ω), x ∈ ∂Ω, and y ∈ Ω, then ∂G(x, y)/∂n is comparable at all points of γ = ∂Ω ∩ D(x, r) +with a constant depending only on an upper bound for +M = max +� +1, +ℓ(γ) +dist(y, ∂Ω) +� +, +where ℓ(γ) denotes the length of γ). Moreover, ∂G(x, y)/∂n = O(1/r) on γ. +Proof. We can cover γ by at most O(M) disks Dj = D(xj, rj) whose doubles are all disjoint +from y. +Lemma 10.6 implies the normal derivatives are comparable with some uniform +constant C on each corresponding arc, so they are comparable with constant CO(M) on γ. +(In fact, if r is so small that ∂Ω looks “straight” on scale r, then only O(log M) disks are +needed, since they become geometrically larger as we move away from y.) +The final claim follows because the integral of the normal derivative over the whole bound- +ary is 2π, and hence the integral over γ is ≤ 2π. Since the size of the normal derivative is +comparable at all points of γ, this implies it is bounded above by O(1/ℓ(γ)) = O(1/r) (recall +ℓ denotes length). +□ +The following two lemmas relate harmonic measure to Green’s function: we refer to Figure +16 for an illustration. + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +35 +(a) +(b) +Figure 16. In (A) the setup for Lemma 10.8 is shown, and in (B) the setup +for Lemma 10.9. +Lemma 10.8. There is a constant C1 < ∞ so the following holds. Suppose Ω is an analytic +domain, that Γ is a connected component of ∂Ω and that w ∈ Ω satisfies +dist(w, Γ) = dist(w, ∂Ω) ≤ diameter(Γ). +Then G(z, w) ≤ C1 on σ = {z : |z − w| = 1 +2 dist(w, Γ)}. +Proof. Let Ω′ be the component of �C \ Γ containing Ω. Then Ω ⊂ Ω′, so by the maximum +principle the Green’s function for Ω is less than the Green’s function for Ω′. Thus it suffices +to prove the lemma for the simply connected domain Ω′. But by Koebe’s distortion theorem, +σ contains a ball of fixed hyperbolic radius around w and hence its image contains a ball of +fixed radius if we conformally map Ω′ to the disk with w going to zero. On the disk, the +Green’s function is log 1 +|z| which is clearly bounded by some C outside a fixed ball around +the origin. +□ +Lemma 10.9. There is a constant C2 < ∞ so that the following holds. Suppose that Ω is +an analytic domain, z ∈ Ω, that γ ⊂ ∂Ω is a subarc, and that w ∈ Ω satisfies +(1) dist(w, Γ) = dist(w, ∂Ω) ≤ diameter(Γ), where Γ is the component of ∂Ω containing +γ, +(2) |z − w| ≥ dist(w, ∂Ω). +Then G(z, w) ≤ C2ω(z, γ, Ω)/ω(w, γ, Ω). +Proof. Let C1 and σ = {z : |z − w| = 1 +2 dist(w, Γ)} be as in Lemma 10.8. By assumption z +is in the component Ω′ of Ω \ σ not containing w and by the maximum principle applied to +Ω′, ω(z, σ, Ω′) ≥ G(z, w)/C1. By the maximum principle (again applied to Ω′), +ω(z, γ, Ω) +≥ +ω(z, σ, Ω′) · min +x∈σ ω(x, γ, Ω). + +r +5 +Wr +Z +W +5 +Y36 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +By Harnack’s inequality (see Theorem 7.17 of [Mar19]), all the values of ω(x, γ, Ω) are +comparable on σ and hence there is an ε > 0 so that +ω(z, γ, Ω) +≥ +ω(z, σ, Ω′) · ε · ω(w, γ, Ω). +Finally, by Lemma 10.8 we have +ω(z, γ, Ω) +≥ +(ε/C1) · G(z, w) · ω(w, γ, Ω), +which is the desired inequality with C2 = C1/ε. +□ +Corollary 10.10. Suppose Ω is an analytic domain, z ∈ Ω and ε > 0. Suppose also that +{γk}n +1 ⊂ ∂Ω is a collection of disjoint arcs and {wk}n +1 ⊂ Ω is a collection of points, so that +for all k we have: +(1) dist(wk, Γk) = dist(wk, ∂Ω) ≤ diameter(Γk), where Γk is the component of ∂Ω con- +taining γk, +(2) |z − wk| ≥ dist(wk, ∂Ω), +(3) ω(wk, γk, Ω) ≥ ε > 0. +Then �n +k=1 G(z, wk) ≤ C2/ε where C2 is the constant from Lemma 10.9. +Proof. By Lemma 10.9, � +k G(z, wk) ≤ (C2/ε) � +k ω(z, γk, Ω) and since the arcs {γk} are +disjoint, we have � +k ω(z, γk, Ω) ≤ 1. +□ +11. Periods of Harmonic Functions +Suppose Ω is an analytic domain with N +1 boundary components Γ0, Γ1, . . . , ΓN, so that +Ω is regular for the Dirichlet problem. Suppose h is harmonic in Ω. In any sub-disk D ⊂ Ω, +h has a harmonic conjugate �h that is well defined up to an additive constant. If γ is a closed +curve in Ω, then we can analytically continue �h along γ until we return to the starting point. +The period of h along γ is the difference between the starting and ending values of �h. If γ is +homologous to a point, the period is zero, but if γ is homologous to a boundary component +of Ω, then the period may be non-zero. The sum of the periods corresponding to all N + 1 +boundary components is always zero (the union of all boundary curves is homologous to zero +in Ω). +Next we consider the periods of certain special functions. For j = 0, . . . , N let ωj be the +harmonic function on Ω that has boundary value 1 on Γj and is 0 on the other boundary +components. Since the boundary components are analytic, each ωj extends to be analytic +across ∂Ω, so the normal and tangential derivatives are well defined, and themselves analytic, +at every boundary point. The period of ωj along Γk is the integral of the tangential derivative +of �ωj around Γk, and this equals the integral of the normal derivative of ωj, i.e., the period +is +λjk = +� +Γk +∂ωj +∂n ds. + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +37 +Note that since 0 < ωj < 1 in Ω and ωj = 1 on Γj, the inward normal derivative of ωj on Γj +is non-positive (and strictly negative by analyticity: in that case, G can’t have critical point +on the boundary). Similarly, the inward normal derivative of ωj is strictly positive on Γk for +k ̸= j. Thus λjj < 0 and λjk > 0 for k ̸= j. Since the periods of ωj sum to zero we have +N +� +k=0 +λjk = 0 for every j, +and since � +j ωj is the constant function 1 on Ω, we have +N +� +j=0 +λjk = 0 for every k. +Thus for each j we have: +λjj = − +� +k≥0:k̸=j +λjk. +In other words, the row and column sums are all zero for the (N +1)×(N +1) matrix (λjk), +0 ≤ j, k ≤ N. +If we drop the first row and column of this matrix, we are removing all positive terms +(except for λ00), so we have j = 1, . . . , N, +λjj < − +� +k≥1;k̸=j +λjk. +Thus the N × N matrix Λ = (λjk), 1 ≤ j, k ≤ N is diagonally dominant, and this implies it +is invertible by the Levy-Desplanques theorem: if the kernel of Λ contains a non-zero vector +v = (v1, . . . , vN), then � +k≥1 vkλjk = 0, and if |vk| is the largest component of v, then +|vkλkk| = | +� +j≥1:j̸=k +λjkvj| ≤ |vk| · +� +j≥1:j̸=k +|λjk| < |vk| · |λkk|, +which is a contradiction. See Olga Taussky-Todd’s paper [Tau49] for some history of this +oft-rediscovered fact. Also note that ∥Λ∥ and ∥Λ−1∥ only depend on Ω. +We have now proven the following result (due in a slightly different form to Heins [Hei50] +and in greater generality to Khavinson [Kha84]). +Lemma 11.1. Suppose Ω, Λ and {ωj}N +1 are as above and suppose we assign real numbers v = +(v1, . . . , vN) to the N boundary components Γ1, . . . , ΓN. Then there is a linear combination +h = �n +j=1 ajωj so that the period of h around Γj is exactly vj for j = 1, . . . , N. +The +coefficients a = (a1, . . . , aN) are solutions of the linear equation Λa = v and hence ∥a∥ ≤ +∥v∥ · ∥Λ−1∥. Thus if ∥v∥ ≤ ε then ∥a∥ = O(ε) with a constant that depends only on Ω. +We say that the periods of a harmonic function h on Ω are well defined modulo 2π if +every period is some integer multiple of 2π. In this case, f = exp(h + i�h) is a well defined + +38 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +analytic function on Ω. For example, if Ω is the complement of a finite set of points {zk}n +1, +then �n +k=1 log |z − zn| has periods that are well defined modulo 2π. +The corresponding +holomorphic function is a polynomial with zeros at {zk} (and hence extends holomorphically +from Ω to the whole plane). Note that ∇�u is always well defined even if �u is not, since any +two different branches of �u differ by a constant. +Corollary 11.2. Suppose Ω is an analytic domain and K ⊂ Ω is a compact set that contains +curves {σj}N +0 homologous to each of the boundary components {Γj}N +0 . Suppose K ⊂ W ⊂ Ω +is open, let η = dist(K, ∂W) and set U = {z ∈ W : dist(z, ∂W) > η/2}. Suppose P ⊂ Ω is a +finite set and suppose u and H are harmonic functions on Ω \ P, and each is either bounded +or has a logarithmic pole at each point of P. Suppose |u − H| is bounded by M on U, and +that |u − H| < ε on K. If u has a well defined harmonic conjugate modulo 2π on Ω, then +there is an harmonic h on Ω so that +(1) h + H also has a well defined harmonic conjugate modulo 2π on Ω \ Z, +(2) |h| ≤ Cεc on all of Ω (not just K), +(3) h is constant on each component of ∂Ω. +The constant c is the same as in Lemma 10.1 and C depends only on Ω and K. +Proof. Since v = u − H is bounded on U, it extends to be harmonic at each point of P ∩ U. +Since |v| < ε on K, it is bounded by O(εcM 1−c) on U and hence the gradient of v is bounded +by O(εcM 1−c/η) on U. Thus the gradient of the (possibly multi-valued) harmonic conjugate +of v is bounded by the same quantity. We deduce that the harmonic conjugates of H and u +differ by δ = O(LεcM 1−c/η) on U, where L is the diameter of U in the path metric. Thus +the periods of H on the {σj} differ from multiples of 2π by at most δ. Now apply Lemma +11.1 to define a harmonic function h on Ω that is bounded by O(δ) and has exactly the +periods of −v. +□ +12. The Generalized Caratheodory Theorem on Blaschke Approximation +If B is a finite Blaschke product (see Definition 2.2) on an analytic domain Ω, then it has +non-zero, continuous boundary values, and hence can have only finitely many zeros inside +Ω. Moreover, the Schwarz reflection principle implies B extends holomorphically across ∂Ω. +The following result extends Carath´eodory’s Theorem (Theorem 2.1), and its statement uses +the notation IB (see Notation 2.4), and the following notation: +Notation 12.1. If Ω ⊂ C is an analytic domain and Γ := ∂Ω, we denote Γδ := {z ∈ Ω : +dist(z, Γ) = δ}. +Theorem 12.2. Suppose that Ω ⊂ C is an analytic domain, K ⊂ Ω is compact, and f is +holomorphic on a neighborhood of Ω with supΩ |f| ≤ 1. Let Γ := ∂Ω. Then for any ε > 0 +and sufficiently small δ > 0, there is a finite Blaschke product B on Ω so that +(1) supK |f − B| ≤ ε, + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +39 +(2) 1 − ε < |B| ≤ 1 on ∂Ω, +(3) Every component of Γδ \ {B = 0} has length comparable to δ and adjacent connected +components have length comparable to within a factor of 1 + ε. +(4) All components of IB have length which is comparable to δ with constants that depend +only on ε and Ω. +(5) on each component γ of IB, the ratio maxγ |B′|/ minγ |B′| is bounded depending only +on ε and Ω. +Proof. Without loss of generality, we may assume K is connected. Otherwise, replace K by +a compact, connected superset, for instance, its closed convex hull in the hyperbolic metric. +As before, let G(z, w) denote the Green’s function on Ω with pole at w. +Since f is holomorphic on a neighborhood of Ω, it only has finitely many zeros in Ω. We +consider g := (1 − a) · f + b for some constants 0 < a < ε and |b| < ε such that |g| < 1 − ε +on Ω and g has no zeros on ∂Ω. If we construct a finite Blaschke product B approximating +g to within ε on K, then it approximates f to within 3ε. Fix a finite number of smooth +curves {σk} that are homologous to each boundary curve of Ω. By enlarging K, if necessary, +we may assume it contains all these curves. +Let {zk}N +1 be the zeros of g, counted with +multiplicity. By enlarging K again, if necessary, we may assume all the zeros of g are in K. +Let W be an open domain with K ⊂ W ⊂ W ⊂ Ω. By compactness we have min∂W |g| ≥ η +for some η > 0. In what follows, δ > 0 will always be chosen so small that W is disjoint +from {z ∈ Ω : dist(z, ∂Ω) < 2δ}. In particular, g has no zeros in this neighborhood of the +boundary. +Let u(z) = − log |g(z)|. Then u ≥ − log(1 − ε) ≥ ε is positive and harmonic on Ω except +for finitely many logarithmic poles at the {zk}N +1 , the zeros of g listed with multiplicity. +Let p(z) = �N +k=1 G(z, zk) be the sum of the Green’s functions with these poles. +Then +v(z) = u(z)−p(z) is harmonic on Ω, and equals u on ∂Ω. Thus v is continuous and non-zero +on ∂Ω, and hence it is bounded and bounded away from zero there, say m ≤ v ≤ M on ∂Ω. +Hence m ≤ v ≤ M on all of Ω by the maximum principle. +By Theorem II.2.5 of [GM08] v is the Poisson integral of its boundary values, i.e., +v(z) = 1 +2π +� +∂Ω +v(w)∂G(w, z) +∂n +ds(w) +where +∂ +∂n is the inward normal and ds denotes length measure on the boundary. For w ∈ Γδ, +denote by w∗ ∈ ∂Ω the closest point to w on ∂Ω. For z ∈ K and w ∈ Γδ we have +G(w, z) = δ · ∂G(w∗, z) +∂n ++ O(δ2), +where the constant depends on z, but is uniformly bounded as long as z is in the compact +set K (the constant in the “big-Oh” depends on a bound for |∇2G| between Γδ and ∂Ω and +since G is harmonic and extends analytically across ∂Ω, this is bounded as long as the pole + +40 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +of Green’s function is not too close to ∂Ω). It follows that +v(z) = +1 +2πδ +� +Γδ +v(w)G(w, z)ds(w) + O(δ). +(12.1) +Next use the identity G(z, w) = G(w, z) (e.g., Theorem II.2.8 of [GM08]), to deduce +v(z) = +1 +2πδ +� +Γδ +v(w)G(z, w)ds(w) + O(δ). +We now discretize the integral by cutting Γδ into disjoint subarcs {γk} chosen so that +1 ≤ +� +γk +v(w) +2πδ ds(w) ≤ 1 + O(δ). +(12.2) +This is possible since the integral over a component Γk +δ of Γδ is at least A = m · ℓ(Γk +δ)/2πδ +and this tends to infinity as δ ↘ 0. We can therefore cut each boundary curve into sub-arcs +where the integral is between 1 and 1+O(1/A) = 1+O(δ), i.e., we can make the sub-integrals +all as close to 1 as we wish, by taking δ small enough. +The left side of Equation (12.2) implies that for each k, we have ℓ(γk)M/2πδ ≥ +� +γk v/2π ≥ +1 and hence ℓ(γk) ≥ 2πδ/M. Similarly, the other side implies ℓ(γk) ≤ (1 + O(δ))2πδ/m. +Thus every such arc has length comparable to δ. If δ is small enough, then the continuity +implies that on the union of two adjacent arcs with common endpoint x, v is close to v(x), +and hence by (12.2), the lengths of these intervals are both close to 2πδ/v(x), and hence are +close to each other. This fact, together with each γk have length comparable to δ, will imply +part (3) of the Theorem once we have defined the generalized Blaschke product B. +Adding and subtracting a term from (12.1), have +v(z) += +1 +2πδ +� +j +G(z, wj) +� +γj +v(w)ds(w) +(12.3) ++ 1 +2πδ +� +j +� +γj +v(w)[G(z, w) − G(z, wj)]ds(w) + O(δ). +The curve Γδ is parallel to the boundary, which is the level line G(z, w) = 0 of the Green’s +function with pole at w. Thus, the gradient of G along Γδ is nearly perpendicular to Γδ. +Denoting by wj the center of γj, we conclude: +|G(z, w) − G(z, wj)| = O(δ2). +(12.4) +Using (12.4) in the last term of Equation 12.3 gives +v(z) += +1 +2πδ +� +j +G(z, wj) +� +γj +v(w)ds(w) + +� +j +1 +2πδ +� +γj +v(w)O(δ2)ds(w) + O(δ). + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +41 +Simplifying, we get +v(z) += +1 +2πδ +� +j +G(z, wj) +� +γj +v(w)ds(w) + O(δ) +� +j +� +γj +v(w)ds(w) + O(δ) += +� +j +G(z, wj) +� +γj +v(w) +2πδ ds(w) + O(δ) += +� +j +G(z, wj)(1 + O(δ)) + O(δ), += +� +j +G(z, wj) + O(δ), +where in the last line we have used Corollary 10.10 to bound the sum of Green’s functions by +O(1). Therefore, v is approximated on K by a finite sum of Green’s functions on Ω (indeed, +we can even take the approximation to hold on the larger compact set W). +If Ω is simply connected, then we are essentially done. In this case, +H(z) += +� +k +G(z, zk) + +� +j +G(z, wj) = p(z) + +� +j +G(z, wj) +is harmonic except for a finite number of logarithmic poles at P = {∪kzk}∪{∪jwj}. Therefore +H has a harmonic conjugate �H that is well defined modulo 2π on Ω \ P. Then +B(z) = exp(−H − i �H) +is holomorphic on Ω \ P, but tends to zero at each point of P, so B is holomorphic on all of +Ω with zeros exactly at the points of P. Moreover, on ∂Ω we have |B| = exp(0) = 1, so B +is a finite Blaschke product on Ω. Moreover, +H(z) = p(z) + +� +j +G(z, wj) = u(z) − v(z) + +� +j +G(z, wj) ≈ u(z) = − log |g(z)|, +so log |B| = −H approximates log |g| on W as closely as wish. In particular, we may assume +|B| ≥ η/2 on ∂W. In this case, g/B is holomorphic on W (since every zero of B inside W is +also a zero of g of the same multiplicity), and so by the maximum principle |g/B| is bounded +on K by max∂K |g/B| ≤ maxK 2|g|/η. Now apply Lemma 10.3 to h = log g/B = u+ �u on W +to deduce that arg(B) approximates arg(g) on K, at least if we add an appropriate constant +to arg(B). Therefore B (times an appropriate unit scalar) uniformly approximates g on K. +This extends Carath´eodory’s theorem to simply connected domains. +However, if Ω is multiply connected, then H need not have a well defined harmonic con- +jugate modulo 2π on Ω \ P. Therefore B = exp(−H − i �H) need not be well defined: if +we analytically continue B along one of the closed loops σk, we return to the same absolute +value, but possibly a different value of the argument. The change in the argument is as small +as we wish, tending to zero as the difference between log |B| = H and log |g| tends to zero + +42 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +on K. This is because g has a well defined harmonic conjugate modulo 2π on Ω \ P, and +so the discussion following Lemma 10.3 applies. In order to get a well defined (generalized) +finite Blaschke product B on Ω, we apply Corollary 11.2 to u and H to construct h so that +(1) h is harmonic on all of Ω, and +(2) h + H has a well defined harmonic conjugate modulo 2π on Ω \ P. +(Note the H was constructed exactly so that the corollary can be applied: u−H is harmonic +except for poles on Γδ which are outside W, and we may make u − H is as close to zero on +K as we wish, while keeping it uniformly bounded on W.) +Now set F = −H − h. This function has all periods equal to zero modulo 2π, so B = +exp(F + i �F) is a well defined holomorphic function on Ω. Since |h| = O(δ), we can deduce +that B still approximates g on the compact set K. Moreover, since H = 0 on ∂Ω and h = aj +on Γj, we see that |B| = exp(aj) = 1 + O(δ) on Γj, so |B| is constant on each boundary +component. Dividing by the largest such value, we get another finite Blaschke product that +satisfies (2) and still approximates g to within O(δ) on K. +To prove (4), it suffices to show that the modulus of the tangential derivative of B along +∂Ω (which is equal to |B′| since B is holomorphic) is comparable to 1/δ everywhere: then +the preimage of a half-circle will have length comparable to δ. Note that on ∂Ω, |B′| is also +equal to the normal derivative of h + H. The function h is a linear combination of fixed +functions ωj that depend only on Ω, and although the coefficients of the combination may +depend on δ, they remain small as δ ↘ 0. Thus the normal derivative of h remains bounded +on ∂Ω as δ ↘ 0, and this is negligible compared to 1/δ. +The function H is a sum of two sets of Green’s functions, one with poles {zj} corresponding +to the zeros of g and the other with poles {wk} along the curve Γδ. The first set of poles is +fixed independent of δ and their contribution to the normal derivative of H is also bounded +independent of δ. Again, these terms are negligible. +The main contribution to the normal derivative of H comes from the poles {wk} lying on +Γδ. For each such point wk consider the arc σk = D(wk, 2δ) ∩ ∂Ω; the harmonic measure of +this arc with respect to wk is bounded uniformly away from zero, i.e., ω(wk, σk, Ω) > c > 0. +This harmonic measure is the integral over σk of the normal derivative of the Green’s function +with pole at wk, and thus it is less than the integral of the normal derivative of H, since this +Green’s function is one term of the sum defining H. Thus the integral of |B′| over σk is > c +and so any single arc lj in IB can contain at most a bounded number of arcs of the form σk. +By Part (3) of the theorem, adjacent points wk are at most distance O(δ) apart and hence +lj can have length at most O(δ) (otherwise it would cover too many of the arcs σk). +Next, we need to prove ℓ(lj) is bounded below by a multiple of δ, and this is equivalent to +proving an upper bound |B′(x)| = O(1/δ) for any x ∈ ∂Ω. As noted above, the contributions +to |B′| from h and from the poles of H coming from the zeros of g are both bounded +independent of δ. To deal with the poles {wk} of H on Γk, we choose a point z ∈ Γδ that is + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +43 +close to x ∈ ∂Ω, say |x − z| ≤ 2δ and note that +∂ +∂nG(x, wk) ≃ 1 +δG(z, wk) +Thus by Corollary 10.10, the total contribution of all the poles {wk} to |B′| is at most +� +k +1 +δG(wk, z) = O(1 +δ), +and hence (4) is proven. +Let γ denote a component of IB. By Corollary 10.7 and conclusion (4) just proven, the +normal derivative of Green’s function with a pole at least distance δ from ∂Ω has comparable +values at all points of γ, and so the same holds for any finite sum of such functions (with +the same constant). Since on ∂Ω we have |B′| = | � ∂G +∂n |, we deduce (5). +□ +Remark 12.3. If Ω = D, then it suffices to assume f is holomorphic on Ω instead of a +neighborhood of Ω. In that case if r < 1 then g(z) = f(rz) is holomorphic on a neighborhood +of D and approximates f on a compact set K ⊂ D if r is close enough to 1. So if B is a finite +Blaschke product on Ω approximating g to within ε/2, and r is chosen so that g approximates +f to within ε/2, then B approximates f to within ε, as desired. +Remark 12.4. The analyticity of f on a neighborhood of Ω is only used to deduce that f +has a finite number of zeros inside Ω. The same proof would work if we assumed that f is +holomorphic on Ω, extends continuously to ∂Ω, and is non-zero on ∂Ω. +Remark 12.5. If we make the previous assumption on f, then it suffices to assume Ω is +bounded and finitely connected. If so, and any component of ∂Ω is a single point p, then +by the Riemann removable singularity theorem, f extends to be holomorphic at p, and +it su���ces the prove the theorem for the extended function on the domain Ω′ = Ω ∪ {p}. +By removing all the point components of ∂Ω, we may assume every component of Ω is +non-trivial. By repeated applications of the Riemann mapping theorem to the complement +of each complementary component of Ω, the domain Ω can be mapped to a domain Ω′ +bounded by a finite number of analytic curves indeed, with more work, the Koebe circle +domain theorem says it can be mapped to a domain bounded by circles. Transferring f and +K to Ω we can use Theorem 12.2 to construct an approximating finite Blaschke product on +Ω′, and then transfer this to a finite Blaschke on Ω that approximates f on K. +Remark 12.6. We placed the poles of our Green’s function all at the same distance δ from +∂Ω, but this was not necessary. If we fix z0 ∈ Ω and let w ∈ Ω approach x ∈ ∂Ω, then +G(z, w)/G(z0) approaches a positive harmonic function on Ω with zero boundary values on +∂Ω, except at x, where it blows up. Thus the limiting function must be a multiple of the +Poisson kernel on Ω with respect to x ∈ ∂Ω. Thus it is easy to find finite weighted sums +of Green’s functions (with positive real weights) that approximate the Poisson integral of v. + +44 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +Then one must cluster the poles to approximate this sum by a sum of Green’s functions with +integral weights; exponentiating such a sum gives a finite Blaschke product on Ω. Possibly +this extra flexibility would be useful in other problems, such trying to minimize the number +of poles needed. +Proof of Theorem 2.6: The hypotheses of Theorem 2.6 are stronger than those of Theorem +12.2: namely we assume in Theorem 2.6 that ||f||Ω < 1 (rather than ≤ 1) and that the +zeros of f are disjoint from ∂Ω. Under these additional assumptions, there is no need in the +second paragraph of the proof of Theorem 12.2 to replace f by g := (1 − a) · f + b since +we already have |f| < 1 − ε and f has no zeros on ∂Ω. Since the B produced in Theorem +12.2 approximates g to within O(δ) and we may take δ → 0, the conclusions of Theorem 2.6 +follow from the conclusions of Theorem 12.2. +□ +References +[ABR01] +Sheldon Axler, Paul Bourdon, and Wade Ramey. Harmonic function theory, volume 137 of Grad- +uate Texts in Mathematics. Springer-Verlag, New York, second edition, 2001. +[Ahl06] +Lars V. Ahlfors. 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Monthly, 56:672–676, 1949. +C.J. Bishop, Mathematics Department, Stony Brook University, Stony Brook, NY 11794- +3651 +Email address: bishop@math.stonybrook.edu +Kirill Lazebnik, Mathematics Department, University of North Texas, Denton, TX, 76205 +Email address: Kirill.Lazebnik@unt.edu +