Closed Jordan Curve Research Articles

Zur Approximation pseudoanalytischer Funktionen durch Pseudopolynome

In the theory of pseudoanalytic functions one can define (pseudoanalytic) rational functions, especially polynomials called “pseudopolynomials”. (See Bers [3], [4], Vekua [12]) Therefore it can be developed a theory of approximation and interpolation by rational functions. First results have been published by Bers [3] (Runge's theorem), Ismailov and Taglieva [8]. Let G be a domain of the complex plane bounded by a closed Jordan curve, let w(z) be pseudoanalytic in G. In this paper we deal with a relation between the behaviour of w(z) on C (Holder-continuity) and the degree of approximation of w(z) by pseudopolynomials. The results correspond to certain theorems of Curtiss, Sewell and Walsh in the theory of analytic functions.

Best uniform approximation by meromorphic functions with free poles

Using some recent results on best polynomial approximations, a method for explicitly calculating the meromorphic function with a fixed number of free poles of best uniform approximation to certain functions on a closed Jordan curve is obtained.

Best Uniform Approximation by Meromorphic Functions with Free Poles

Using some recent results on best polynomial approximations, a method for explicitly calculating the meromorphic function with a fixed number of free poles of best uniform approximation to certain functions on a closed Jordan curve is obtained.

Fundamental constants for rational functions

Suppose R is a rational function with n poles all of which lie inside Γ \Gamma , a closed Jordan curve. Lower bounds for the uniform norm of the difference R − p R - p on Γ \Gamma , where p is any polynomial, are obtained (in terms of the norm of R on Γ \Gamma ). In some cases these bounds are independent of Γ \Gamma as well as R and p. Some related results are also given.

On the Convergence of Best Uniform Deviations

If a function f is continuous on a closed Jordan curve $\Gamma$ and meromorphic inside $\Gamma$, then the polynomials of best uniform approximation to f on $\Gamma$ converge interior to $\Gamma$. Furthermore, the limit function can in each case be explicitly determined in terms of the mapping function for the interior of $\Gamma$. Applications and generalizations of this result are also given.

Approximation by $\delta $-Polynomials

The approximation to complex-valued functions, continuous on a closed Jordan curve by polynomials of degree n, whose uniform norm on that curve is greater than or equal to some prescribed constant, is investigated. The limits for the resultant sequence of the best such deviations are found for a large class of functions.

Ein verbesserter Existenzsatz f�r Fl�chen konstanter mittlerer Kr�mmung

Let r be a recifiable closed Jordan curve in the euclidean 3-space IR3, and denote by Ar the infimum of the areas of all surfaces bounded by r. Then for every real number H with\(\left| H \right| \leqslant 0.52 \cdot \sqrt \pi /\sqrt {A_\Gamma }\) we show the existence of a surface with boundary curve r having constant mean curvature H (except in possible branching points). This improves a theorem of WENTE. Given an isolated minimal surface bounded by r for sufficiently small |H| we further prove the existence of a surface of constant mean curvature with boundary curve r which is close to the minimal surface.

A characterization of badly approximable functions

Complex valued functions, continuous on a closed Jordan curve in the plane and having the property that all of their polynomials of best uniform approximation on that curve are identically zero are characterized in terms of their mapping properties on that curve.

Grunsky inequalities and coefficients of bounded schlicht functions

Abstract : Let D be a plane finitely connected domain whose boundaries are closed Jordan curves C sub nu and let f(z) be a regular analytic function in D whose coefficients in power series expansion about a point zeta epsilon D are b sub nu. Grunsky had given a set of necessary and sufficient conditions, depending upon b sub nu, so that f(z) may be extended as a schlicht function in D. In the present report, Grunsky's conditions have been extended to the case when f(z) has the additional restriction of being bounded, i.e., /f(z)/ less than 1, z epsilon D. These conditions have then been used to get distortion theorems and coefficient inequalities for bounded schlicht functions in /z/ less than 1 and in /z/ greater than 1. (Author)

On the distortion in conformal mapping of variable domains

Formanyproblems inconformal mapping it is of importance to have precise quantitative information concerning the variation of the mapping function of a region onto a circle, under a deformation of the region. A number of estimates for this variation in terms of parameters, which measure the deformation and the smoothness of the boundary of the region, have been given in the literature. A survey of such results is contained in [13] and [i1 ](2). While the degree of the change in the mapping function itself has been investigated for various general configurations, the corresponding question concerning the distortion, i.e. the derivative of the mapping function, has been studied primarily for nearly circular regions. The object of the present paper is to obtain such estimates of the distortion for arbitrary regions under quite general hypotheses. The paper is divided into two parts. The question considered in the first part arose in connection with the following theorem [8, p. 364]. Suppose C and Cn (n= 1, 2, * * * ) denote closed Jordan curves which are represented by the equations

Some Fixed Point Theorems

where f, p are continuous, g satisfies a Lipschitz condition, p(t) has period 1, and g(t)/I ? 1 for large t at any rate. Our choice of hypotheses and the main lines of our investigations have been dominated by what is significant in the theory of differential equations, but our results are concerned solely with sets of points and transformations of sets of points. In the first place if a solution of (1) for which o = 0o, = v = no1, when t= 0 has t = 0i, vq = ni, at t = 1, then the transformation 3 of the point x = (6 77o) on to the point xi = (4i, nj) is (1, 1) and continuous, and also orientation preserving in the open I, -q Cartesian plane. That is to say as the point x describes a closed Jordan curve J counter-clockwise, the point xi = 3(x) describes J1 = 3(J) counter clock-wise. Even when we consider closed invariant subsets of the plane, we continue to assume that 3 is (1, 1) continuous and orientation-preserving in the whole plane and not merely in the subset considered; for the tranformations set up by the solutions of the differential equation in which we are especially interested always satisfy these hypotheses. We use methods depending essentially on these hypotheses, and although there may be generalizations of some of our results of one kind or another, we do not attempt to discuss them. We shall also give sotne special consideration to transformations which decrease areas or leave them the same size, as this type of result is easily verified for transformations defined by certain classes of such differential equations. A complete and accurate statement of our aims and results would need various lengthy definitions and the introduction of new notation, but we shall first describe the main lines we have followed, and our reasons for doing so, in terms which to a large extent explain themselves. Precise definitions of the terms used will be given later in the appropriate places. We are indebted to Prof. L. C. Young and to Mr. H. D. Ursell for many helpful criticisms and suggestions.

Distortion Theorems in the Theory of Schlicht Functions

Let be the family of analytic functions F(z) which are regular and schlicht in the circle E: |z| < 1 and normalized at the origin such that F(0) = 0 and F′(0) = 1. Let be the subfamily of consisting of all functions, each of which possesses, as boundary of its image-domain., a closed Jordan curve which is supposed to be regular analytic. Then is everywhere dense in the original family . Hence, by an approximation theorem of Carathéodory on the kernel of sequence of domains, we may restrict ourselves within , so far as we concern the estimation of continuous functionals on .

On Several Families of Plane Curves with an Application to Differential Equations

1. Let D be a domain bounded by a closed Jordan curve J. Let us assume that a family F of simple continuous curves is defined in the domain, with the conditions that 1: Through every point of D and of its boundary passes one and only one curve of the family. (The curve through the point P will be designated by C(P).) 2: No curve of the family has an end point in the interior of D. THEOREM. The curve C(P), regarded as a function of P, is continuous; that is to say, if P. is a sequence of points converging to the point P, the curves C(P(F) converge uniformly to the curve C(P). PROOF. Let us suppose our theorem false. Then, there exists a positive number e and a sequence Pni contained in the sequence PR, such that each curve C(Pni) has a point pi whose distance from the curve C(P) is greater than C. Let p be a limit point of the set pi. This point p is not on the curve C(P). From the sequence C(Pi) we can extract a new sequence such that the corresponding points pi converge to p. Then, successively extracting new sequences of curves from this sequence, each sequence contained in the preceding, and using the well known method of the diagonal, we can obtain a sequence of curves C* which possesses the following property: If E is the set of the points x such that every neighborhood of x has points in common with an infinite number of the curves C*, then, if 5 is a given neighborhood of x, there exists a number N such that every curve C*, with n > N, has points in S. The set E is obviously a bounded continuum joining P and p. Now, let us divide the points of E into disjoint sets E, in such a manner that two points of E belong to E, if and only if they belong to a same curve of the family F. Every set E,, being the intersection of two closed sets (the set E and a curve of the family F), is closed. As is well known,' a bounded continuum cannot be divided into a finite number (greater than 1), or into a countable number of closed disjoint sets. Therefore, if we prove that the sets E. are not uncountable, it will follow that there exists only one set E,. Then the set E would be an arc of a curve C, of the family F, joining P and p, and consequently distinct from the curve C(P) which does not contain the point p. But this is impossible, since C(P) is the only curve of the family F that passes through P.