Adaptive signal and system approximation and strong divergence

Abstract

Many divergence results for sampling series are in terms of the limit superior and not the limit. This leaves the possibility of a convergent subsequence. If there exists a convergent subsequence, adaptive signal processing techniques can be used. In this paper we study sampling-based signal reconstruction and system approximation processes for the space PW π 1 of bandlimited signals with absolutely integrable Fourier transform. For all analyzed examples, which include the peak value of the Shannon and the conjugated Shannon sampling series, we prove strong divergence, i.e., divergence for all subsequences. Hence, adaptive signal processing techniques do not help in these cases. We further analyze whether an adaptive choice of the reconstruction functions in the oversampling case can improve the behavior.