Strong asymptotics for Bergman polynomials over domains with corners and applications

TL;DR

This paper derives strong asymptotics for Bergman polynomials over domains with corners using quasiconformal mapping theory, extending classical results and enabling various applications in complex analysis and operator theory.

Contribution

It introduces a new approach based on quasiconformal mappings to establish asymptotics for domains with corners, filling a gap in classical Bergman polynomial theory.

Findings

Established strong asymptotics for Bergman polynomials on domains with corners.

Applied results to coefficient estimates in univalent function classes.

Connected asymptotics to operator theory, capacity computation, and moment-based reconstruction.

Abstract

We establish the strong asymptotics for Bergman orthogonal polynomials defined over Jordan domains with corners. This complements an investigation started in 1923 by T. Carleman, who derived the strong asymptotics for domains bounded by analytic curves, and carried over by P.K. Suetin in the 1960's, who established them for domains with smooth boundaries. In order to do so, we use a new approach based on tools from quasiconformal mapping theory. The impact of the resulting theory is demonstrated in a number of applications, varying from coefficient estimates in the well-known class Sigma of univalent functions and a connection with operator theory, to the computation of capacities and a reconstruction algorithm from moments.