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Title page for ETD etd-06062006-132316


Type of Document Dissertation
Author Mina Diaz, Erwin
Author's Email Address erwin.d.mina@vanderbilt.edu
URN etd-06062006-132316
Title Asymptotics for Faber polynomials and polynomials orthogonal over regions in the complex plane
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Edward B. Saff Committee Chair
Akram Aldroubi Committee Member
Dechao Zheng Committee Member
Douglas P. Hardin Committee Member
Prodyot K. Basu Committee Member
Keywords
  • Bergman Kernel
  • Carleman Theorem
  • Faber Polynomials
  • Orthogonal Polynomials
  • Zeros of Polynomials
Date of Defense 2006-05-16
Availability unrestricted
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 $|w(z)|^2dm(z)$.

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