Bergman Orthogonal Polynomials and the Grunsky Matrix

Abstract

By exploiting a link between Bergman orthogonal polynomials and the Grunsky matrix, probably first observed by Kühnau (Ann Acad Sci Math 10:313–329, 1985), we improve on some recent results on strong asymptotics of Bergman polynomials outside the domain G of orthogonality, and on the 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.