Random Sampling and Reconstruction of Non‐Decaying Signals From Weighted Multiply Generated Shift‐Invariant Spaces

ABSTRACT It is a classical assumption in sampling theory that the input signal is square‐integrable. However, non‐decaying signals are prevalent in science and engineering. They do not decay and even grow at infinity, such as realizations of Brownian and Lévy processes. In this paper, we consider the problem of random sampling of non‐decaying signals from weighted multiply generated shift‐invariant spaces. We obtain that with overwhelming probability, the random sampling stability holds uniformly for all non‐decaying signals in certain compact subsets of the weighted multiply generated shift‐invariant spaces when the sampling size is sufficiently large. Furthermore, we consider a reconstruction algorithm for a subspace of the weighted multiply generated shift‐invariant spaces and provide the probability that the algorithm succeeds.