Approximation of conformal mapping via the Szeg\H{o} kernel method

TL;DR

This paper investigates how well polynomials derived from Szeg\

Contribution

It introduces a new approximation method for conformal mappings using Szeg\

Findings

Convergence is uniform on the closure of Smirnov domains.

Approximation rate depends on the smallest exterior angle.

Inside the domain, convergence rate is quadratic relative to boundary convergence.

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\H{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.