Overconvergence of Series and Potential Theory

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.