Asymptotics for Hessenberg Matrices for the Bergman Shift Operator on Jordan Regions

Abstract

Let G be a bounded Jordan domain in the complex plane. The Bergman polynomials {pn}n=0∞ of G are the orthonormal polynomials with respect to the area measure over G. They are uniquely defined by the entries of an infinite upper Hessenberg matrix M. This matrix represents the Bergman shift operator of G. The main purpose of the paper is to describe and analyze a close relation between M and the Toeplitz matrix with symbol the normalized conformal map of the exterior of the unit circle onto the complement of Ḡ. Our results are based on the strong asymptotics of pn. As an application, we describe and analyze an algorithm for recovering the shape of G from its area moments. © 2012 Springer Basel AG.