Asymptotics for Hessenberg matrices for the Bergman shift operator on Jordan regions
Provides theoretical insights into Bergman shift operators and a practical algorithm for shape recovery, but the results are incremental for experts in complex analysis and orthogonal polynomials.
The paper establishes a relation between the Hessenberg matrix for Bergman orthogonal polynomials and a Toeplitz matrix, using strong asymptotics, and applies this to recover the shape of a Jordan domain from its area moments.
Let G be a bounded Jordan domain in the complex plane and consider the infinite upper Hessenberg matrix M associated with the Bergman orthogonal polynomials of G. This matrix represents the Bergman shift operator of G. The main purpose of the paper is to describe and analyze a close relation between M and the Toeplitz matrix with symbol the normalized conformal map of the exterior of the unit circle onto the complement of the closure of G. Our results are based on the strong asymptotics of the Bergman polynomials. As an application, we describe and analyze an algorithm for recovering the shape of G from its area moments.