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., $G^star:=Gsetminus mathcal{K}$. 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.
Download