Bergman orthogonal polynomials and the Grunsky matrix

TL;DR

This paper explores the connection between Bergman orthogonal polynomials and the Grunsky matrix to improve asymptotic results and analyze boundary regularity, introducing a new matrix approach for quasiconformal boundaries.

Contribution

It introduces a novel matrix method linking Bergman polynomials and the Grunsky matrix, enhancing understanding of asymptotics and boundary regularity.

Findings

Improved asymptotic estimates for Bergman polynomials outside the domain.

New insights into the entries of the Bergman shift operator.

Application of the approach to quasiconformal boundaries.

Abstract

By exploiting a link between Bergman orthogonal polynomials and the Grunsky matrix, probably first observed by K{\"u}hnau in 1985, we improve some recent results on strong asymptotics of Bergman polynomials outside the domain G of orthogonality, and on entries of the Bergman shift operator. In our proofs we suggest a new matrix approach involving the Grunsky matrix, and use well-established results in the literature relating properties of the Grunsky matrix to the regularity of the boundary of G, and the associated conformal maps. For quasiconformal boundaries, this approach allows for new insights for Bergman polynomials.