Strong divergence of the Shannon sampling series for an infinite dimensional signal space

Abstract

Knowing whether a reconstruction process, for example the Shannon sampling series, is strongly divergent in terms of the lim or only weakly divergent in terms of the lim sup is important, because strong divergence is linked to the non-existence of adaptive reconstruction processes. For non-adaptive reconstruction processes the existence is answered by the Banach-Steinhaus theory. However, the analysis of adaptive reconstruction processes is more difficult and not covered by the former theory. In this paper we consider the Paley-Wiener space PW π 1 of bandlimited signals with absolutely integrable Fourier transform and analyze the structure of the set of signals for which the peak value of the Shannon sampling series is strongly divergent. We show that this set is lineable, i.e., that there exists an infinite dimensional subspace, all signals of which, except the zero signal, lead to strong divergence. Consequently, for all signals from this subspace, adaptivity in the number of samples that are used in the Shannon sampling series does not create a convergent reconstruction process.