Singularities of Analytic Functions and Group Representations

Abstract

In this thesis, we demonstrate some connections between the coefficients of a Taylor series $f(z)=\sum_{n=0}^\infty a_n z^n$ and singularities of the function. There are many known results of this type; for example, counting the number of poles on the circle of convergence and doing convergence or overconvergence for $f$ on any arc of holomorphy. Here, a new approach is proposed in which these kinds of results are extended by relaxing the classical conditions for singularities and convergence theorems. This is done by allowing the coefficients to be sufficiently small instead of being zero. The well-known theta function is an important example. Every point on the boundary of its domain of holomorphy is singular. This function is delivered from the \emph{covariant transform} associated with the Heisenberg group representations. Therefore, we devote the rest of our present work to deal primarily with the \emph{covariant transform}. We introduce three different forms of the Heisenberg group representations. The \emph{covariant transform} allows us to construct intertwining operators related to \(\FSpace{L}{2}\)-type spaces between the representations of the Heisenberg group. The systematic usage of the covariant transform between different spaces on which the Heisenberg group representations act is another new contribution in this thesis.