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Overconvergence of Series and Potential Theory
Author(s)
Date Issued
2021
Date Available
2022-04-29T13:27:24Z
Abstract
Let f be a holomorphic function on a domain W in the complex plane, where W contains the unit disc D. Suppose that a subsequence of the partial sums of the Taylor expansion of f about 0 is locally uniformly bounded on a subset E of the complex plane. Then, depending on the nature of E, it may be possible to infer additional information about the convergence of the subsequence on W. If E is non-thin at infinity, then the subsequence converges locally uniformly to f on W. If E is non-polar and does not meet the boundary of D, then the subsequence converges locally uniformly to f on a neighborhood of every point z on the boundary of D such that the complement of W is thin at z. In this thesis we consider similar phenomena in other settings. In Chapter 4 we investigate properties of harmonic homogeneous polynomial expansions of harmonic functions on R^N and use complexification along real lines to obtain analogues for the above results. Let h be harmonic on a domain W in R^N. First, we show that, if a subsequence of the partial sums of the expansion of h is locally uniformly bounded on a sequence of balls with certain properties, then this subsequence converges to h on W. Surprisingly, this sequence of balls may be thin at infinity in higher dimensions. Second, suppose that W contains the unit ball and a subsequence of the partial sums of the expansion of h about 0 is locally uniformly bounded on a ball of radius greater than 1. Then this subsequence of the partial sums converges on a neighborhood of every regular point of h on the boundary of the unit ball. We apply these results to questions of existence of universal polynomial expansions of harmonic functions. In Chapter 5 we study universal Laurent expansions of harmonic functions. In Chapter 6 we study subsequences of Dirichlet series. In this case the analogy with Taylor series is closer, but a new aspect is the role played by the Martin boundary and minimal thinness.
Type of Material
Doctoral Thesis
Publisher
University College Dublin. School of Mathematics and Statistics
Qualification Name
Ph.D.
Copyright (Published Version)
2021 the Author
Language
English
Status of Item
Peer reviewed
This item is made available under a Creative Commons License
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