Orthogonal Polynomials for Area-Type Measures and Image Recovery

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.