# Approximation of Conformal Mapping via the Szegő Kernel Method

Approximation of Conformal Mapping via the Szegő Kernel Method 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 Szegő 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

# Approximation of Conformal Mapping via the Szegő Kernel Method

, Volume 3 (1) – Mar 1, 2004
16 pages

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Publisher
Springer Journals
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/bf03321026
Publisher site
See Article on Publisher Site

### 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 Szegő 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.

### Journal

Computational Methods and Function TheorySpringer Journals

Published: Mar 1, 2004

Keywords: Conformal mapping; Szegő kernel; Fourier series; orthogonal polynomials; 30C40; 30E10; 41A10; 30C30

### References

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