#### 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 *R ^{p}(X)* as the closure of the rational functions with poles off

*X*in

*L*. It is well known that for 1 ≤

^{p}(X, dA)*p*< 2,

*R*=

^{p}(X)*L*. Although density may not be achieved for

^{p}(X)*p*> 2, there exists a set

*X*so that

*R*=

^{p}(X)*L*for

^{p}(X)*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

*R*lies almost everywhere on the set of bounded point evaluations of

^{p}(X)*X*.

#### Recommended Citation

Mattingly, Christopher, "RATIONAL APPROXIMATION ON COMPACT NOWHERE DENSE SETS" (2012). *Theses and Dissertations--Mathematics*. 4.

https://uknowledge.uky.edu/math_etds/4