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(star) obtained from G by removing K; i.e., G(star) := G \ K. We refer to G as an archipelago and G(star) as an archipelago with lakes. Denote by {p(n)(G, z)}(n=0)(infinity) and {p(n)(G(star), z)}(n=0)(infinity) the sequences of the Bergman polynomials associated with G and G(star), respectively, that is, the orthonormal polynomials with respect to the area measure on G and G(star). The purpose of the paper is to show that p(n)(G, z) and p(n)(G(star), z) have comparable asymptotic properties, thereby demonstrating that the asymptotic properties of the Bergman polynomials for G(star) are determined by the boundary of G. As a consequence we can analyze certain asymptotic properties of p(n)(G(star), z) by using the corresponding results for p(n)(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.