Approximation of Conformal Mappings Approximation of Conformal Mappings of Annular Regionsy

In this paper we examine the convergence rates in an adaptive version of an orthonormalization method for approximating the conformal map ping f of an annular region A onto a circular annulus In particular we consider the case where f has an analytic extension in compl 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 AMS classi cation C E A Introduction Let A be an annular region of the complex plane bounded by two piecewise analytic Jordan curves e and i where i is interior to e and denote the exterior of e by so that and the interior of i by G Also assume that G let z be a xed point on i and denote by f the conformal mapping of A onto a circular annulus E fw jwj Mg so that f z Here the outer radiusM of E is uniquely determined by A and is called the conformal module of A The work of this paper is concerned with certain aspects of the conver gence theory of an orthonormalization method ONM which was studied by Papamichael et al and for the purpose of approximating the conformal mapping f and the conformal moduleM The details of the basic ONM are as follows Let L A be the Hilbert space of all square integrable functions that are analytic in A with inner product