Strong asymptotics for Bergman polynomials over domains with corners and applications
For mathematicians working in complex analysis and approximation theory, this provides the first rigorous asymptotics for Bergman polynomials on domains with corners, filling a long-standing gap since Carleman (1923) and Suetin (1960s).
The paper establishes strong asymptotics for Bergman orthogonal polynomials over Jordan domains with corners, extending prior results from analytic and smooth boundaries. The new theory is applied to coefficient estimates in univalent functions, operator theory, capacity computation, and moment-based reconstruction.
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.