Approximation of Analytic Functions by Faber Polynomials, the Grunsky Matrix, and a Univalence Criterion

Abstract: The aim of this thesis is derive a set of polynomials defined on simply connected domains, the Faberpolynomials, in which all analytic function on the domain can be uniformly approximated. Importantconcepts and theorems such as isomorphisms, automorphisms and the Riemann mapping theorem areintroduced. Examples and applications are also included. Furthermore, the thesis will aim to introducean important consequence of the Faber polynomials, the method of the Grunsky inequalities. The first section introduces important properties of analytic functions and the concept of isomor-phisms, in particular the form of all automorphisms of the unit disc will be derived. The second sectionconsiders the Riemann mapping theorem, a theorem that relates any simply connected region that is notall of ℂ to the unit disc. A proof of the theorem beginning with the Arzelá-Ascoli theorem is provided.An application in constructing harmonic functions on arbitrary simply connected regions will be pre-sented. In the third section, definitions and properties of the Faber polynomials are developed; followedby simple examples. The section concludes with a proof and example of the statement that analyticfunctions can be approximated by Faber polynomials. In the fourth and last section of the thesis, themethod of Grunsky inequalities is presented. Starting off, the Grunsky coefficients are defined using theFaber polynomials. Properties of Grunsky coefficients such as the symmetry property and the Grunskyinequalities are then derived. To conclude it will be shown that the Grunsky inequalities provide aunivalence criterion for analytic functions defined on the unit disc.