Year of Publication

2012

Degree Name

Doctor of Philosophy (PhD)

Document Type

Doctoral Dissertation

College

Arts and Sciences

Department

Mathematics

First Advisor

Dr. James E. Brennan

Abstract

For a compact, nowhere dense set X in the complex plane, C, define Rp(X) as the closure of the rational functions with poles off X in Lp(X, dA). It is well known that for 1 ≤ p < 2, Rp(X) = Lp(X) . Although density may not be achieved for p > 2, there exists a set X so that Rp(X) = Lp(X) for p up to a given number greater than 2 but not after. Additionally, when p > 2 we shall establish that the support of the annihiliating and representing measures for Rp(X) lies almost everywhere on the set of bounded point evaluations of X.

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