Some Basic Convergence Results

Let us now consider the convergence of DP Fourier series and of the corresponding complex power series. In many cases the convergence of the complex power series, which is the Taylor series of the inner analytic function, and which is usually more easily established, will give us information about the convergence of the DP Fourier series. However, sometimes the reverse situation may also be realized. For organizational reasons this discussion must be separated into several parts. We will tackle in this paper only the extreme cases which we will qualify as those of strong divergence and strong convergence, for which results can be established in full generality. Further discussions of the convergence issues will be presented in the follow-up paper [1].