Fine asymptotics for Bergman polynomials over domains with corners
Provides the first rigorous asymptotics for Bergman polynomials over domains with corners, filling a gap in approximation theory for non-smooth boundaries.
The paper derives fine asymptotic formulas for the leading coefficients and Bergman polynomials in the complement of a bounded simply-connected domain with corners, extending prior results for analytic and smooth boundaries.
We consider the series of the Bergman orthogonal polynomials associated with a bounded simply-connected domain in the complex plane, whose boundary is a Jordan curve. These are the polynomials that are orthonormal with respect to the area measure on the domain. The purpose of this note is to report on recent results regarding the fine asymptotic behaviour of the the leading coefficients and the polynomials in the complement of the domain, in cases when the boundary includes corners. These results complement an investigation started in the 1920's by T. Carleman, who obtained the fine asymptotics for domains with analytic boundaries and carried over by P.K. Suetin in the 1960's, who established them for domains with smooth boundaries.