The thesis belongs to the field of complex analysis, which deals with the boundary behaviour of power series. The goal of the thesis is to analyse the connection between the sequence of coefficients in a power series with the possibility of holomorphic extension of the function defined locally by the series to the boundary points of the convergence domain of the series. This is attempted by studying lacunary series, which are power series with consecutive blocks of zero coefficients called gaps. Several criteria are derived that indicate under what conditions lacunary series cannot be extended to any boundary point. Furthermore, the concept of overconvergence is defined, which is closely related to the gaps in lacunary series. The thesis concludes with a proof of Szegö’s extension theorem.
|
