Global and Local Approximation Behavior of Reconstruction Processes for Paley-Wiener Functions

For signals in the Paley-Wiener space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$PW_\pi ^1$$\end{document} a reconstruction in the form of a sampling series that is uniformly convergent on compact subsets of the real axis and uniformly bounded on the whole real axis is not possible in general if the signals are sampled equidistantly at Nyquist rate. We prove that, even if the signal is non-uniformly sampled with an average sampling rate equal to the Nyquist rate, a uniformly convergent reconstruction is not possible. Additionally, we provide a detailed convergence analysis and give a sufficient condition for the uniform convergence of the Shannon sampling series without oversampling. However, if oversampling is applied, a uniformly convergent reconstruction is always possible and as far as convergence is concerned no elaborate kernel design is necessary. Moreover, we show that a projection of the reconstruction process onto the range of signal frequencies is not possible without losing the good convergence behavior.