Local and global convergence behavior of non-equidistant sampling series

In this paper we analyze the local and global convergence behavior of sampling series with non-equidistant sampling points for the Paley-Wiener space PW_pi ¹ and sampling patterns that are made of the zeros of sine-type functions. It is proven that the sampling series are locally uniformly convergent if no oversampling is used and globally uniformly convergent if oversampling is used. Furthermore, we show that oversampling is indeed necessary for global uniform convergence, because for every sampling pattern there exists a signal such that the peak value of the approximation error grows arbitrarily large if no oversampling is used. Finally, we use these findings to obtain similar results for the mean-square convergence behavior of sampling series for bandlimited wide-sense stationary stochastic processes.