Klaus Menke

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] (Runges 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 (Hölder-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.

On tsuji points in a continuum

Let and E be a compact subset of D.Tsuji [4] introduced a point system on E by which the hyperbolic capacity ρ(E) of E can be approximated. If E is a continuum, the point system can also be used to approximate the conformal mapping of D\E onto (see Pommerenke [1], Siciak [3]). Here some sharper estimates for the approximations are given under additional assumptions on E.

Point systems with extremal properties and conformal mapping

SummaryLetD be the unit disk. It is a well-known fact that by use of “simply connected domain methods” the general conformal mapping problem of doubly connected domains can be reduced to the special case of a regionD bounded by the unit circle and a Jordan curve γ inD, where

Lösung des Dirichlet-Problems bei Jordangebieten mit analytischem Rand durch Interpolation

0 \notin \bar D

Some properties of functions schlicht in an annulus

. Here we treat this special case and assume γ to be piecewise analytic without cusps. Let Φ be the conformal mapping of {ρ<|w|<1} onto the doubly connected domainD with Φ(1)=1. We approximate Φ by interpolation with finite Laurent series using point systems with extremal properties. Numerical results for four examples are given.

Conformal Mapping of Doubly Connected Regions

LetC be a closed Jordan curve in the complex plane and letf(z)=dz+a0+a1z−1+… be the analytic function mapping |z|>1 schlicht onto the exterior ofC (d>0 is the transfinite diameter ofC). Similar to the Fekete points a point system will be defined calledextremal points. One can use the Fekete points or the extremal points to approximated. The author has proved [3] that in the case of an analytic closed Jordan curve the approximation ofd by means of extremal points is much better than the approximation ofd by the use of Fekete points. Here we show how to approximated by means of extremal points in the case of a piecewise analytic, closed Jordan curve possessing corners of openingαπ (0

Coefficient estimates for functions univalent in an annulus

AbstractLetC be an analytic Jordan curve, letG be the interior ofC, and letU (w) be (at least) continuous onC. Here the solution of the Dirichlet problemu(w) which coincides withU(w) onC is approximated by harmonic polynomials. These harmonic polynomialsFnF(w) are determined by interpolatingU(w) in a given point system. For sufficiently greatn we prove |u(w)−Fn(w)|≤K·logn·En in