Bergman orthogonal polynomials and the Grunsky matrix
Classical Analysis and ODEs
2016-07-07 v2 Complex Variables
Functional Analysis
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.
Cite
@article{arxiv.1606.00553,
title = {Bergman orthogonal polynomials and the Grunsky matrix},
author = {Bernhard Beckermann and Nikos Stylianopoulos},
journal= {arXiv preprint arXiv:1606.00553},
year = {2016}
}