Approximation of Conformal Mapping via the Szegő Kernel Method

Abstract

We study the uniform approximation of the canonical conformal mapping, for a Jordan domain onto the unit disk, by polynomials generated from the partial sums of the Szeg˝ o kernel expansion. These polynomials converge to the conformal mapping uniformly on the closure of any Smirnov domain. We prove estimates for the rate of such convergence on domains with piecewise analytic boundaries, expressed through the smallest exterior angle at the boundary. Furthermore, we show that the rate of approximation on compact subsets inside the domain is essentially the square of that on the closure. Two standard applications to the rate of decay for the contour orthogonal polynomials inside the domain, and to the rate of locally uniform convergence of Fourier series are also given. 1. Convergence of the Szeg˝ o kernel expansion and approximation of conformal maps Let G be a Jordan domain in the complex plane. There are two well known kernel methods used for approximation of the canonical conformal mappings of G onto a disk. The Bergman kernel method is associated with the L 2 spaces and orthogonal polynomials with respect to the area measure, while the Szeg˝ o kernel method is based on the inner product and orthogonal polynomials with respect to the arclength measure on the boundary of G (see Gaier [7], Smirnov and Lebe-dev [19]). The approximations related to the area orthogonality approach were

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