Computational Methods and Function Theory 3 (2003), No. 1, 79--94
Copyright Heldermann Verlag 2003
Igor E. Pritsker
igor@math.okstate.edu, Department of Mathematics, 401 Mathematical Sciences, Oklahoma State University, Stillwater, OK 74078-1058, U.S.A.
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 Szegö 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.
Keywords: Conformal mapping, Szegö kernel, Fourier series, orthogonal polynomials.
MSC 2000: Primary 30C40, 30E10; Secondary 41A10, 30C30.
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