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 $\mathcal{K}$ be a compact subset of $G$ and consider the set $G^{\star }$ obtained from $G$ by removing $\mathcal{K}$; i.e., $G^{\star }:=G\setminus \mathcal{K}$. We refer to $G$ as an archipelago and $G^{\star }$ as an archipelago with lakes. Denote by $\{p_n(G,z){\}}_{n=0}^{\infty }$ and $\{p_n(G^{\star },z){\}}_{n=0}^{\infty }$, 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.

24 pages, 9 figures