Orthogonal polynomials for area-type measures and image recovery

Saff, E. B. and Stahl†, H. and Stylianopoulos, N. and Totik, Vilmos (2015) Orthogonal polynomials for area-type measures and image recovery. SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 47 (3). pp. 2442-2463. ISSN 0036-1410

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Abstract

Let G be a finite union of disjoint and bounded Jordan do- mains 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 or- thonormal 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 as- ymptotic properties of the Bergman polynomials for G⋆ are determined by the boundary of G. As a consequence we can analyze certain as- ymptotic 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 recon- struction algorithm for recovering the shape of an archipelago with lakes from a partial set of its complex moments.

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