Approximation of conformal mappings of annular regions

Abstract

In this paper we examine the convergence rates in an adaptive version of an orthonormalization method for approximating the conformal mapping \(f\) of an annular region \(A\) onto a circular annulus. In particular, we consider the case where \(f\) has an analytic extension in compl(\(\overline{A}\)) and, for this case, we determine optimal ray sequences of approximants that give the best possible geometric rate of uniform convergence. We also estimate the rate of uniform convergence in the case where the annular region \(A\) has piecewise analytic boundary without cusps. In both cases we also give the corresponding rates for the approximations to the conformal module of \(A\).