Asymptotics for Faber polynomials and polynomials orthogonal over regions in the complex plane

Abstract

In this dissertation we first study the Faber polynomials for a piecewise analytic Jordan curve $L$ without inner cusps (some extra conditions are additionally imposed on $L$). Let $Omega$ and $G$ be, repectively, the exterior and interior domains of $L$. We obtain uniform asymptotics for these polynomials holding on any closed subset of $Omegacup L$ without nonsmooth corners, and on any compact set contained in $G$. We also derive fine statements on the zeros of these polynomials. Secondly, we study polynomials that are orthogonal over the interior $G$ of a Jordan curve $L$ with respect to a measure of the form $|w(z)|^2dm(z)$, where $w otequiv 0$ is an analytic function on $G$ and $m$ is the area measure. When $L$ is analytic and $wequiv 1$, we derive an integral representation for these polynomials that allows us to obtain strong type of asymptotics holding inside the curve $L$ and from which fine statements on the zeros of the polynomias follow. For a general $w$ we obtain results that relate the zero distribution of the orthogonal polynomials with the singularities of the reproducing kernel of the space of all analytic functions on $G$ that are square integrable with respect to $|w(z)|^2dm(z)$.