dc.creator | Mina Diaz, Erwin | |
dc.date.accessioned | 2020-08-22T17:03:28Z | |
dc.date.available | 2007-06-09 | |
dc.date.issued | 2006-06-09 | |
dc.identifier.uri | https://etd.library.vanderbilt.edu/etd-06062006-132316 | |
dc.identifier.uri | http://hdl.handle.net/1803/12483 | |
dc.description.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 $|w(z)|^2dm(z)$. | |
dc.format.mimetype | application/pdf | |
dc.subject | Bergman Kernel | |
dc.subject | Carleman Theorem | |
dc.subject | Faber Polynomials | |
dc.subject | Orthogonal Polynomials | |
dc.subject | Zeros of Polynomials | |
dc.title | Asymptotics for Faber polynomials and polynomials orthogonal over regions in the complex plane | |
dc.type | dissertation | |
dc.contributor.committeeMember | Prodyot K. Basu | |
dc.contributor.committeeMember | Akram Aldroubi | |
dc.contributor.committeeMember | Douglas P. Hardin | |
dc.contributor.committeeMember | Dechao Zheng | |
dc.type.material | text | |
thesis.degree.name | PHD | |
thesis.degree.level | dissertation | |
thesis.degree.discipline | Mathematics | |
thesis.degree.grantor | Vanderbilt University | |
local.embargo.terms | 2007-06-09 | |
local.embargo.lift | 2007-06-09 | |
dc.contributor.committeeChair | Edward B. Saff | |