Bergman polynomials on an Archipelago: Estimates, Zeros and Shape Reconstruction

TL;DR

This paper investigates the asymptotic behavior of Bergman polynomials on complex archipelagos, revealing geometric insights, and develops a shape reconstruction algorithm based on orthogonal polynomial asymptotics and moments.

Contribution

It provides new growth estimates, analyzes zero distributions, and introduces a shape reconstruction method for archipelagos using orthogonal polynomial asymptotics.

Findings

Asymptotic zero distribution reflects archipelago geometry.

Explicit asymptotics for lemniscate archipelagos are derived.

Shape reconstruction from finitely many moments is achieved.

Abstract

Growth estimates of complex orthogonal polynomials with respect to the area measure supported by a disjoint union of planar Jordan domains (called, in short, an archipelago) are obtained by a combination of methods of potential theory and rational approximation theory. The study of the asymptotic behavior of the roots of these polynomials reveals a surprisingly rich geometry, which reflects three characteristics: the relative position of an island in the archipelago, the analytic continuation picture of the Schwarz function of every individual boundary and the singular points of the exterior Green function. By way of explicit example, fine asymptotics are obtained for the lemniscate archipelago $|z^m-1|<r^m, 0<r<1,$ which consists of $m$ islands. The asymptotic analysis of the Christoffel functions associated to the same orthogonal polynomials leads to a very accurate reconstruction algorithm of the shape of the archipelago, knowing only finitely many of its power moments. This work naturally complements a 1969 study by H. Widom of Szeg\H{o} orthogonal polynomials on an archipelago and the more recent asymptotic analysis of Bergman orthogonal polynomials unveiled by the last two authors and their collaborators.