Approximation of Cauchy-Type Integrals

The classical Green's function associated to a simply connected domain in the complex plane is easily expressed in terms of a Riemann mapping function. The purpose of this paper is to express the Green's function of a finitely connected domain in the plane in terms of a single Ahlfors mapping of the domain, which is a proper holomorphic mapping of the domain onto the unit disc that is the analogue of the Riemann map in the multiply connected setting.

Let G and {αkj} be the domain and the array mentioned in the title (the boundary of the domain is assumed to be rectifiable). One describes a general scheme for the approximation of fonctionsf in the domain G, representable in the form f(z)=(2πi)−1∫g(ς)(ς−z)−1dς, where g ∈ Lz (∂G), by a sequence of rational fractions. The characteristic feature of this scheme is the fact that the poles τk of the fraction lie in the k-th row of the array {αkj}. There is given a condition on {αkj}, necessary and sufficient in order that each functionf, of the kind described above, should admit a uniform approximation inside G with the aid of the indicated scheme. In the case when this condition is not satisfied and ℂ\G.

This paper is devoted to an investigation of a topological ring of analytic functions. Specifically, this ring, denoted by R, is the set of functions analytic on the unit disc with the usual addition and scalar multiplication, the Hadamard product for its ring multiplication, and the compact-open topology. The ring R is identified algebrai cally with a subring RA of the ring of continuous functions on the non-negative integers X. The operations in fT are the usual pointwise operations, ana the structure of R is determined by considering its isomorph iT. In Chapter I we are concerned with the problems of identifying the maximal ideal space of R and describing the maximal ideals intrinsically. We first show, using theorems on general rings of continuous functions, that the maximal ideals are in one-to-one correspondence with the points of the Stone-Cech compactification px of X. We next give an intrinsic description of the maximal ideals, using the properties of the power series expansions for analytic functions. Using this description we strengthen the prev ious theorem appreciably and show that the maximal ideal space with the hul1-kernel topology is homeomorphic to pX. Finally, the Hadamard product is used to give a simple iv characterization of the dual space of the topological linear space of analytic functions on the unit disc. This dual space is isomorphic to the set of functions in R whose radius of convergence exceeds one, which is exactly the intersection of the maximal ideals corresponding to points of pX -X (the dense maximal ideals of R) . In Chapter II we continue the investigation of the maximal ideals by studying the structure of their associated residue class rings. The complex number field 0 is isomorphically embedded in R/M, where M is a maximal ideal of R. If M corresponds to a point of X, then R/M and the isomorph 0* of 0 are identical; whereas, if M corresponds to a point of ^X -X, then R/M is a transcendental extension of 0* having transcendence degree c, the cardinality of the continuum. Moreover, we show, in the second case, that R/M is algebraically closed. Using theorems on transcendental extensions and algebraically closed fields, we show that, in either case, R/M and 0 are isomorphic fields. The two classes of maximal ideals are distinguished by the fact that their residue class rings admit radically different types of complex-valued isomorphisms. In Chapter III we are concerned primarily with the structure of the closed ideals of R. The basic tool used is the rotational completeness theorem for analytic v functions, which we proved using the methods and results of harmonic analysis. We show that the closure of every principal ideal is principal, give a necessary and suffi cient condition that a principal ideal be closed, and show that every closed ideal is a principal ideal generated by an idempotent element of R. Using these theorems we indicate connections with the general theory of dual rings, of which R is an example, and raise several questions for further i investigation in the direction of releasing some of the restrictions with which most of the results so far have been obtained.

In a recent paper the first author gave an explicit construction of a sequence of polynomials having their zeros on the unit circumference which converge boundedly to a given bounded zero-free analytic function in the unit disk. In this paper we find the best possible uniform bound for such approximating polynomials and construct a sequence for which this bound is attained. The method is also applied to approximation of an analytic function in the unit disk by rational functions whose poles lie on the unit circumference. Some open problems are discussed.

CONTENTS Introduction Chapter I. The analytic capacity of sets § 1. Definition and some properties of analytic capacity § 2. The connection between the capacity of a set and measures § 3. On removable singularities of analytic functions § 4. The analytic C-capacity of sets § 5. Estimates of the coefficients in the Laurent series § 6. The change in the capacity under a conformal transformation of a set Chapter II. The separation of singularities of functions § 1. The construction of a special system of partitions of unity § 2. Integral representations of continuous functions § 3. Separation of singularities § 4. The approximation of functions in parts § 5. Approximation of functions on sets with empty inner boundary § 6. The additivity of capacity for some special partitions of a set Chapter III. Estimation of the Cauchy integral § 1. Statement of the result § 2. Estimate of the Cauchy integral § 3. Estimation of the Cauchy integral along a smooth Lyapunov curve § 4. Proof of the principal theorem § 5. Some consequences § 6. A refinement of the Maximum Principle and the capacity analogue to the theorem on density points Chapter IV. Classification of functions admitting an approximation by rational fractions § 1. Examples of functions that cannot be approximated by rational fractions § 2. A criterion for the approximability of a function § 3. Properties of the second coefficient in the Laurent series § 4. Proof of the principal lemma § 5. Proof of the theorems of § 2 Chapter V. The approximation problem for classes of functions § 1. Removal of the poles of approximating functions from the domain of analyticity of the function being approximated § 2. Necessary conditions for the algebras to coincide § 3. A criterion for the equality of the algebras § 4. Geometrical examples § 5. Some problems in the theory of approximation Chapter VI. The approximation of functions on nowhere dense sets § 1. The instability of capacity § 2. A capacity criterion for the approximability of functions on nowhere dense sets § 3. Theorems on the approximation of continuous functions in terms of Banach algebras § 4. A capacity characterization of the Mergelyan function and of peak points References

This paper concerns a certain subalgebra of the Banach algebra of complex valued, essentially bounded, Lebesgue measurable functions on the unit circle in the complex plane (denoted here by L∞). My interest in this subalgebra was prompted by a question of R. G. Douglas. Let H∞ denote the space of functions in L∞ whose Fourier coefficients with negative indices vanish (equivalently, the space of boundary functions for bounded analytic functions in the unit disk). Douglas [5] has asked whether every closed subalgebra of L∞ containing H∞ is determined by the functions in H∞ that it makes invertible. More precisely, is such an algebra generated by H∞ and the inverses of the functions in H∞ that are invertible in the algebra? An affirmative answer is known for L∞ itself and for certain subalgebras of L∞ recently studied by Davie, Gamelin, and Garnett [3]. At the time of this writing, no algebra is known for which the above question can be answered negatively.

In normed spaces of functions analytic in the Jordan domain Ω⊂ℂ, we establish exact order estimates for the Kolmogorov widths of classes of functions that can be represented in Ω by Cauchy-type integrals along Γ = ∂Ω with densities f(·) such that \(f \circ \Psi \in L_{\beta ,p}^\Psi (T)\). Here, Ψ is a conformal mapping of \(C\backslash \overline \Omega \) onto {w: |w| > 1}, and \(L_{\beta ,p}^\Psi (T)\) is a certain subset of infinitely differentiable functions on T = {w: |w| = 1}.

In this paper we construct concrete convolution polynomials for which quantitative results in approximation of functions in Bergman spaces on the unit disk are obtained, with the estimates expressed in terms of higher order \(L^{p}\)-moduli of smoothness and in terms of the \(L^{p}\)-best approximation quantity, \(0<p<+\infty \).

Let Ω be a domain in the complex plane \(\mathbb{C}\) containing the origin and let \({\cal P}_{n}^{0}(\Omega)\) denote the set of all polynomials P of degree ≤ n satisfying the conditions \(P(0)=0\) and \(P(\mathbb{D})\subset \Omega\), where \(\mathbb{D}=\left\{z\,:\,\,|z|<1\right\}\) is the unit disk in the complex plane. The maximal range, denoted by Ω n , is then defined as the union of all sets \(P(\mathbb{D}), P \in {\cal P}_{n}^{0}\) .We shall construct extremal polynomials and maximal polynomial range for a domain of intersection of two circular disks.

Let G be a n-connected domain in the complex plane and let z0 ∈ G be fixed. We consider the class H1(G, z0) of all holomorphic functions f of G into the unit disk E with f(z0) = 0. For z ∈ Gz0 the Caratheodory distance c(z, z0) is defined as the maximum of ¦f(z)¦, f ∈ H1(G, z0). We determine explicitely the balls z ∈ G: c(z, z0) < r in the case n = 2. Grunsky proved that the extremal functions with respect to the Caratheodory distance are proper mappings of degree n. Here we add that only in the case n = 2 each proper mapping in H1(G, z0) is somewhere in G extremal.

INTRODUCTION. This note concerns Riemann mappings of rectangles onto the unit disk in the complex plane. Let's fix some notation for the discussion. For the unit circle and unit disk we'll use, respectively, S1 and D. For the rectangle parallel to the coordinate axes, centered at the origin 0 E C with height 1 and base length a > 0, so that the corners are at (?a/2, ? 1/2), we'll use R(a). In particular, it will be convenient to have R(b) c R(a) if b < a. Riemann's mapping theorem says that every simply connected plane domain Ql not equal to the whole plane can be mapped conformally one-to-one onto D. Thus this certainly holds for the interior of rectangles. These maps are thoroughly well understood from several points of view, primarily using the Schwarz-Christoffel formula (elliptic functions in a not too subtle disguise). The purpose of this note is to show that rectangles with different modulus (different values of a) can be seen as conformally different and completely characterized by sets of four distinct points on S with very little machinery. In fact, our only tools will be the Schwarz reflection principle, the Schwarz lemma (in other words, Schwarz seemed to have a pretty good sense about this sort of thing), some elementary conformal mappings, and the concept of normal families of analytic functions. Perhaps then the real purpose of this note is to serve as a reminder of the power of the profound, beautiful, and essentially elementary maximum modulus principle. The only reference necessary for this discussion is [A]. The theorem of Lowner may also be found in [C]. The proof given there is of a somewhat different flavor, and Lowner's proof [L] was roughly the same as the one given here.

In the case p = 1 the question remains open: Given an open Riemann surface E does there exist a proper holomorphic embedding of E into C2? Trying to answer this question it is natural to begin with planar domains, i.e. open connected subsets of C. Very few results are known and even in the simplest cases the construction of such an embedding is not easy. Kasahara and Nishino [KN, St] used a technique involving the Fatou-Bieberbach map from C 2 to C 2 to prove that the unit disc can be properly holomorphically embedded into C 2. Their method can be used to prove that for each M E ,W there is an M-connected domain in C which can be properly holomorphically embedded into C 2. Laufer [La] showed that every annulus can be properly holomorphically embedded into C 2. Alexander [AI] used elliptic modular functions to construct such an embedding of the punctured disc into C 2 which, after a slight modification, becomes a proper holomorphic embedding of the unit disc into C 2.

A plane domain G w , symmetric with respect to the origin O, is extended by “sector reflections” to a symmetric domain Ğ w . A “mapping function” [Wbreve](z) of Ğ w (mapping the unit disk G z onto Ğ w ) is constructed in an elementary way in terms of a mapping function w(z) of G w using essentially the reflection principle. In the case of polygonal domains considered here, the results can be interpreted as relations between Schwarz-Christoffel integrals. This method applies to most domains considered in [2]. where only the ratio of conformal radii Ro(Ğ w )/R o (G w ) was determined. The study of the mapping functions themselves was suggested by Max Schiffer.

We study the maximal number 0 • h • +1 for a given plane domain › such that f 2 H p whenever p < h and f is analytic in the unit disk with values in ›: One of our main contributions is an estimate of h for unbounded K-quasidisks.

Convex and alpha-prestarlike subordination chains