Strong Asymptotics for Bergman Polynomials over Domains with Corners and Applications

Abstract

Let G be a bounded simply-connected domain in the complex plane ℂ, whose boundary Γ {colon equals} ∂G is a Jordan curve, and let {pn}n=0 ∞denote the sequence of Bergman polynomials of G. This is defined as the unique sequence pn(z) = λnzn +..., λn> 0, n =0,1,2,..., of polynomials that are orthonormal with respect to the inner product 〈f,g〉 {colon equals} ∫Gf(z)g(z)̄d A (z), where d A stands for the area measure. We establish the strong asymptotics for pnand λn, n∈ℕ, under the assumption that Γ is piecewise analytic. This complements an investigation started in 1923 by T. Carleman, who derived the strong asymptotics for Γ analytic, and carried over by P.K. Suetin in the 1960s, who established them for smooth Γ. 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 Σ of univalent functions and a connection with operator theory, to the computation of capacities and a reconstruction algorithm from moments. © 2012 Springer Science+Business Media New York.