Characterization of the pointwise and the peak value behavior of system approximation under thresholding

In this paper we study the system approximation process, which naturally emerges when a stable linear time-invariant (LTI) system is applied on the Shannon sampling series, for the case that the samples of the signal are disturbed by the non-linear threshold operator, which sets all samples that are below some threshold to zero. We analyze its behavior for signals in the Paley-Wiener space PW¹_π of bandlimited signals with absolutely integrable Fourier transform as the threshold tends to zero. We treat the pointwise as well as the global behavior and characterize the systems for which there exist signals such that the approximation error diverges as the threshold tends to zero. We further show that for those systems in a certain topological sense almost all signals lead to divergence and that the divergence can be arbitrarily fast.