Analytic Jordan Curves Research Articles

Domains bounded by analytic Jordan curves

I t is well known that repeated application of the Riemann mapping theorem shows that any finitely connected plane domain is conformally equivalent to a domain whose boundary components are analytic curves. A corresponding result has not previously been shown for arbitrary domains of infinite connectivity—the only cases being those domains for which the Koebe conjecture is known to be true (cf. [2]). We will show that any domain with countably many boundary components is conjormally equivalent to a domain bounded by analytic Jordan curves and points.

On unicity of capacity functions

Sario's capacity function of a closed subset γ of the ideal boundary is known to be unique if γ is of positive capacity. The present paper will determine the number of capacity functions of γ in terms of the Heins harmonic dimension when γ has zero capacity, under the assumption that γ is isolated. This includes the special case where γ is the ideal boundary. 1* Capacity functions* Denote by β the ideal boundary of an open Riemann surface R in the sense of Kerekjartό-Stoϊlow. We consider a fixed nonempty closed subset 7 c β which is isolated from S — β — 7. Throughout this paper D will denote a fixed parametric disk about a fixed point ζ e R with a fixed local parameter z and the uniqueness is always referred to this fixed triple (ζ, D, z). Here we do not exclude the case where 7 = β. For a regular region Ω Z) D we denote by yΩ the part of dΩ which is homologous to 7. The remainder δΩ = dΩ — yΩ consists of a finite number of analytic Jordan curves δΩj. For a regular exhaustion {RfΛn^o with Ro => D and nonempty τΛo, set yn = ΎE% and δni = dRnJ. Then there exists a unique function py% e H(Rn — ζ) satisfying (a) pΐn I D = log I z - ζ I + hn(z) with hn e H(D) and Λw(ζ) = 0, (b) pϊn I 7n = fcnCr) (const.) and pϊn δnj = d ni (const.) so that %dpr% = 0, which is called a capacity function of jn (Sario [6]). It is known that kn(y) increases with n and the limit k(y) is independent of the choice of {i2J*= 0. We call β~fc(r) the capacity of 7 and denote it by cap 7. When cap 7 > 0, pTn converges to a functions pr, which

Harmonic differential with prescribed singularities

1. Throughout this paper we denote by R an open Riemann surface and by Ro a relatively compact subdomain of R with the relative boundary aRo consisting of a finite number of mutually disjoint closed analytic Jordan curves. The open set R, = R Po can be considered to be a neighborhood of the ideal boundary 3 of R. For the sake of simplicity, we denote by a the common relative boundary aRo = aR and we fix the orientation of a positively with respect to the domain Ro. A harmonic differential a defined on R1 = R1 u a is called a harmonic singularity

On semi-parabolic Riemann surfaces

On finite surfaces, the class of harmonic functions which are constant on each contour is a finite-dimensional vector space of functions with finite Dirichlet norm. This paper considers the corresponding class of functions on bordered surfaces of class SOg and generalizes some of the properties of harmonic measures on finite surfaces. In particular, for generalized harmonic measures, we investigate the level curves and their orthogonal trajectories. The principal results, Theorems 4.1 and 4.4, state, in a sense made precise, that almost all of the level curves of a generalized harmonic measure are analytic Jordan curves and almost all of their orthogonal trajectories begin and end on the border given in the definition of the surface. These results have application to the level curves of a Green's function via a theorem of Kuramochi. We also consider the question on a parabolic surface as to when a harmonic differential with finite norm and integral periods is a weak limit of period reproducing differentials.

On harmonic continuation

Let D be a bounded, closed, simply-connected domain whose boundary C consists of a finite number of analytic Jordan curves. Let γ be any analytic arc of C. Then we shall prove the following theorem.Theorem 1. Let u(x, y) be harmonic in the interior of D and continuous on γ, and let ϱu(x, y)/ϱn=g(s) when (x, y) is on γ, where g(s) is an analytic function of arc-length s along γ. Then u(x, y) can be harmonically continued across γ.

On canonical conformal maps of multiply connected domains

In a recent paper [II], J. L. Walsh has established the existence of two new canonical domains for the conformal mapping of multiply connected regions. The canonical maps may be characterized by the fact that they transform certain functions into functions which are extendible harmonically to the entire plane, with the exception of a finite number of points, one in each of the components of the complement of the canonical domain. Two other existence proofs have been given, by H. Grunsky [6] and J. A. Jenkins [7]. All of these proofs rely on uniformization. In part I of the present paper we give a geometric and constructive proof of existence, based primarily on potential-theoretic considerations. In part II we consider the relation between the new canonical mapping functions and extremal problems, and show thereby that the maps are natural generalizations of those which take, respectively, a doubly connected onto an annulus and simply connected onto the unit circle. Finally, in part III, by specializing the existence proof to the case of a doubly connected domain, we obtain a new construction procedure for the conformal map of such a onto an annulus, and show it to converge more rapidly than one commonly used at present. The author is deeply grateful to Professors J. L. Walsh, L. V. Ahlfors, and R. Osserman for their inspiration and help. I. Existence of the canonical map. Throughout this discussion let the term harmonic measure of an annular domain be taken to mean the measure of the outer boundary curve with respect to the domain; that is, the function which is in the and takes the values 0, 1 on the inner and outer boundary curves respectively. Also, let the term contour denote an analytic Jordan curve. THEOREM 1. Let C1 be a Jordan curve in the z-plane, and D its exterior. Let

Integrated continuity conditions and degree of approximation by polynomials or by bounded analytic functions

Introduction. In the study of uniform approximation to a function of a complex variable by polynomials or by bounded analytic functions, Lipschitz conditions have proved extremely useful in relating degree of approximation to continuity properties of the functions approximated. Parts of this theory are analogous to the older study (S. Bernstein, D. Jackson, de la Vallee Poussin, Montel) of approximation to real periodic functions by trigonometric sums, where Lipschitz conditions have also proved useful. Hardy and Littlewood [2, p. 633 ](2) first pointed out that trigonometric approximation in the mean is likewise closely related to integrated Lipschitz conditions; proofs of the theorems stated by Hardy and Littlewood were first published by Quade [l]. The object of the present note is to indicate rapidly and with a minimum of detail, that degree of approximation in the mean by polynomials in the complex variable and by bounded analytic functions is also conveniently investigated by use of integrated Lipschitz conditions. The investigation leads naturally to the use also of classes of analytic functions satisfying integrated Zygmund and integral asymptotic conditions. We shall approximate functions on an analytic Jordan curve or set of curves in the z-plane. Throughout the paper 7 denotes the unit circle, C a single analytic Jordan curve or a finite number X of mutually exterior analytic Jordan curves Cy, j=l, 2, • • , X; if X = l, Ci = C. A function z = Xi(w), = reie, maps the closed interior of y: \w =1 onto the closed interior of Cy one-to-one and conformally; = fl,(z) denotes the function inverse to z = Xy(-0; Cy denotes the interior of Cy together with Cy. Similarly, = cp(z) maps the exterior of C conformally (not necessarily one-to-one) onto | w > 1 so that 00 = 1) is the image in the exterior of C ol \w =R under this mapping. In this paper, the letter s will denote arc-length measured on C, the letter p a number not less than unity, the letter k a non-negative integer unless otherwise indicated, the letters M and L with or without subscript constants independent of re, z and w. A function f(z) belongs to the class Hp on y if it is analytic interior to y and if Mp[f(rew)}={fln\f(reiB)Yd6}1'r> is bounded for 0<r<l. A function f(z)

Approximation on a line segment by bounded analytic functions: Problem 𝛽

Problem ,B is the study of degree of approximation on a closed bounded point set E to a function f(z) analytic on E, by functions analytic and bounded in a region D containing E; here f(z) is supposed not analytic throughout D, but to possess certain continuity properties on the boundary of a suitable region of analyticity, which contains E and whose closure lies in D. This is a problem on which considerable progress has recently been made [1] if E is bounded by analytic Jordan curves. It is our present purpose to indicate further progress, now for E a line segment. We prove the

On canonical conformal maps of multiply connected regions

In a recent paper [II], J. L. Walsh has established the existence of two new canonical domains for the conformal mapping of multiply connected regions. The canonical maps may be characterized by the fact that they transform certain harmonic domain functions into functions which are extendible harmonically to the entire plane, with the exception of a finite number of points, one in each of the components of the complement of the canonical domain. Two other existence proofs have been given, by H. Grunsky [6] and J. A. Jenkins [7]. All of these proofs rely on uniformization. In part I of the present paper we give a geometric and constructive proof of existence, based primarily on potential-theoretic considerations. In part II we consider the relation between the new canonical mapping functions and extremal problems, and show thereby that the maps are natural generalizations of those which take, respectively, a doubly connected domain onto an annulus and simply connected domain onto the unit circle. Finally, in part III, by specializing the existence proof to the case of a doubly connected domain, we obtain a new construction procedure for the conformal map of such a domain onto an annulus, and show it to converge more rapidly than one commonly used at present. The author is deeply grateful to Professors J. L. Walsh, L. V. Ahlfors, and R. Osserman for their inspiration and help. I. Existence of the canonical map. Throughout this discussion let the term "harmonic measure of an annular domain" be taken to mean the harmonic measure of the outer boundary curve with respect to the domain; that is, the function which is harmonic in the domain and takes the values 0, 1 on the inner and outer boundary curves respectively. Also, let the term "contour" denote an analytic Jordan curve. THEOREM 1. Let C1 be a Jordan curve in the z-plane, and D its exterior. Let

Minimizing operators on subregions

For constructing harmonic functions with a prescribed local behavior, an operator method on arbitrary Riemann surfaces was recently introduced by the author [1]. We showed the existence of a normal linear operator minimizing the Dirichlet integral and referred to other operators to be given later. In the present paper, a general class of minimizing operators will be introduced, including the above operator as a special case. In the existence proof, use will be made of the extremal method presented in [21. Let R be an arbitrary Riemann surface'and G a subregion, compact or not, of finite or infinite genus, relatively bounded by a finite set a of closed analytic Jordan curves. Let v be a real single-valued function on a, harmonic in an open set containing ax. A normal linear operator L in G is defined [1 ] as follows. With every v on a is associated, by L, a unique single-valued harmonic function Lv on G which satisfies the following conditions:

On Interpolation and Approximation by Functions Analytic and Bounded in a Given Region

Introduction. Several new problems in the theory of approximation to an analytic function were suggested and discussed by Walsh in a recent paper(1). It is the purpose of the present paper to make additional contributions to the solution of these problems and to indicate the extent to which the results carry over to the theory of harmonic functions. 1. Representation of a special harmonic function. Let R be a finite sum of disjoint regions in the complex plane and let S be a closed set interior to R. When suitable restrictions are placed upon the boundaries Co and C, of S and R, respectively, there exists a function /(z) equal to zero on Co and unity on C,, continuous in the extended complex plane, harmonic except upon Co and Ci. It is convenient for the present to assume that R and S satisfy the following set of conditions: Al. The boundary of each component region T of R consists of a finite number of disjoint analytic Jordan curves. A2. For each component T, the boundary of TS consists of a finite number of analytic Jordan curves such that those constituting the boundary of each component region of T -TS are disjoint. A3. Each component T contains at least one point of S. A4. No point of TTS is separated from S by the boundary of T. Under these conditions, the function +(z) exists, is necessarily equal to zero on S and unity exterior to R. Let it(z) be a function conjugate to 4(z) in R-S. The limit of it(z), as z, remaining within a component region of R -S, approaches a curve of Co or C1, exists in the small and forms on the boundary of this component region a continuous periodic function of the arc length. Of course, if a component curve of Co or Ci is common to the boundaries of two such component regions of R-S, the limits of the function it(z) from the two sides of this curve are, in general, not equal.

On the Polynomial Derivative Constant for an Ellipse

Let Pn(z) be a polynomial* of degree n in z = x+iy and let j P (z) j < M on a set E, where M is a constant independent of n and z. The author has shownt that if the set E is bounded by an analytic Jordan curve C then j P ' (z)1 < K(C) MAln, where K(C) is a constant depending only on C. If C is the unit circle we know by a theorem of M. Rieszt that K(C) = 1. Here we will show that if C is an ellipse with semi-axes a and b. a _ b, then K(C) ? 1/b. Let us suppose that the ellipse C has its vertices in the points a and -a on the real axis. Then for a point z = x +iy on C we have x = a cos 0, y = b sin 0, and