Strong asymptotics for Bergman polynomials over domains with corners and applications

Details Strong asymptotics for Bergman polynomials over domains with corners and applications Strong asymptotics for Bergman polynomials over domains with corners and applications

2012-02-05

arxiv.org/abs/1202.0994v1

Mathematics

Complex Variables

40 pages, 3 figures. arXiv admin note: substantial text overlap with arXiv:0910.1788

Scientific paper

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.

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