An impossibility result for linear signal processing under thresholding

In this paper, we analyze the approximation of the outputs of linear time-invariant systems by sampling series that use only the samples of the input signal. The samples are disturbed by the threshold operator, which sets all samples with an absolute value smaller than some threshold to zero. We do the analysis for the space of Paley-Wiener signals with absolutely integrable Fourier transform and show for the Hilbert transform that the peak approximation error can grow arbitrarily large for some signals in this space when the threshold approaches zero. This behavior is counterintuitive because one would expect a better behavior if the threshold was decreased. Since we consider oversampling and all kernels from a certain meaningful set, the results are valid not only for one specific approximation process, but for a whole class of approximation processes. Furthermore, we give a game theoretic interpretation of the problem in the setting of a game against nature and show that nature has a universal strategy to win this game.