Orthogonal polynomials for area-type measures and image recovery

Abstract

Let G be a finite union of disjoint and bounded Jordan domains in the complex plane, let K be a compact subset of G, and consider the set G∗ obtained from G by removing K

i.e., G∗ := G \ K. We refer to G as an archipelago and G∗ as an archipelago with lakes. Denote by {pn(G, z)}∞n=0 and {pn(G∗, z)}∞n=0 the sequences of the Bergman polynomials associated with G and G∗, respectively, that is, the orthonormal polynomials with respect to the area measure on G and G∗. The purpose of the paper is to show that pn(G, z) and pn(G∗, z) have comparable asymptotic properties, thereby demonstrating that the asymptotic properties of the Bergman polynomials for G∗ are determined by the boundary of G. As a consequence we can analyze certain asymptotic properties of pn(G∗, z) by using the corresponding results for pn(G, z), which were obtained in a recent work by B. Gustafsson, M. Putinar, and two of the present authors. The results lead to a reconstruction algorithm for recovering the shape of an archipelago with lakes from a partial set of its complex moments. © 2015 Society for Industrial and Applied Mathematics.