Strong asymptotics for Bergman polynomials over non-smooth domains
Let \(G\) be a bounded simply-connected domain in the complex plane \(\mathbb{C}\) whose boundary \(\Gamma := \partial G\) is a Jordan curve and let \(\{ p_n\}_{n=0}^\infty \) denote the sequence of Bergman polynomials of \(G\), that is, the sequence of polynomials \(p_n(z)= \lambda_n z^n + \cdots\), where \(\lambda_n >0\) and \(n=0,1,2,\dots\), that are orthonormal with respect to the inner product \(<f,g> = \int_G f(z) \overline{g(z)} \,dA(z)\), where \(dA\) stands for the area measure. The aim of this note is to report on the strong asymptotics of the leading coefficients \(\{ \lambda_n \}\) and the Bergman polynomials \(\{ p_n\}\) in the case when the boundary \(\Gamma \) is piecewise analytic. The results complement an investigation started by \textit{T. Carleman} [Ark. för Mat., Astron. och Fys. 17, No. 9, 215--244 (1923; JFM 49.0708.03)] under the assumption that \(\Gamma \) is analytic, and continued in the 1960's by P. K. Suetin for smooth \(\Gamma\). As applications, the author shows how the results on strong asymptotics can be used in order to refine a classical and some recent results in the theory of complex orthogonal polynomials.
scientific article; zbMATH DE number 5669087
scientific article