Let <i>K</i> be a compact subset of complex <i>N</i>-dimensional space, where <i>N</i> ≥ 1. Let <i>H</i>(<i>K</i>) denote the functions analytic in a neighborhood of <i>K</i>. Let <i>R</i>(<i>K</i>) denote the closure of <i>H</i>(<i>K</i>) in <i>C</i>(<i>K</i>). We study the problem: What is <i>R</i>(<i>K</i>)?
The study of <i>R</i>(<i>K</i>) is contained in the field of rational approximation. In a set of lecture notes, T. Gamelin [6] has shown a certain operator to be essential to the study of rational approximation. We study properties of this operator.
Now let <i>K</i> be a compact subset of real <i>N</i>-dimensional space, where <i>N</i> ≥ 2. Let harm<i>K</i> denote those functions harmonic in a neighborhood of <i>K</i>. Let <i>h</i>(<i>K</i>) denote the closure of harm<i>K</i> in <i>C</i>(<i>K</i>). We also study the problem: What is <i>h</i>(<i>K</i>)?
The study of <i>h</i>(<i>K</i>) is contained in the field of harmonic approximation. As well as obtaining harmonic analogues to our results in rational approximation, we also produce a harmonic analogue to the operator studied in Gamelin's notes. / Ph. D.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/39825 |
| Date | 13 October 2005 |
| Creators | Ferry, John |
| Contributors | Mathematics, Olin, Robert F., McCoy, Robert A., Rossi, John F., Thomson, James E., Wheeler, Robert L. |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Language | English |
| Detected Language | English |
| Type | Dissertation, Text |
| Format | vi, 164, BTD, application/pdf, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
| Relation | OCLC# 24707058, LD5655.V856_1991.F477.pdf |
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