Fine asymptotics for Bergman polynomials over domains with corners

TL;DR

This paper investigates the detailed asymptotic behavior of Bergman polynomials and their leading coefficients for domains with corners, extending classical results from smooth and analytic boundaries to more complex geometries.

Contribution

It provides new asymptotic results for Bergman polynomials over domains with corners, filling a gap in the classical theory that focused on smooth or analytic boundaries.

Findings

Asymptotic formulas for leading coefficients in domains with corners

Behavior of Bergman polynomials in the complement of such domains

Extension of classical asymptotic results to non-smooth boundaries

Abstract

We consider the series of the Bergman orthogonal polynomials associated with a bounded simply-connected domain in the complex plane, whose boundary is a Jordan curve. These are the polynomials that are orthonormal with respect to the area measure on the domain. The purpose of this note is to report on recent results regarding the fine asymptotic behaviour of the the leading coefficients and the polynomials in the complement of the domain, in cases when the boundary includes corners. These results complement an investigation started in the 1920's by T. Carleman, who obtained the fine asymptotics for domains with analytic boundaries and carried over by P.K. Suetin in the 1960's, who established them for domains with smooth boundaries.