Relevant sampling in finitely generated shift-invariant spaces

We consider random sampling in finitely generated shift-invariant spaces V(Phi) subset of L-2(R-n) generated by a vector Phi = (phi(1,...,)phi(r)) is an element of (L-2(R-n))(r). Following the approach introduced by Bass and GrOchenig, we consider certain relatively compact subsets V-R,(delta)(Phi) of such a space, defined in terms of a concentration inequality with respect to a cube with side lengths R. Under very mild assumptions on the generators, we show that for R sufficiently large, taking O(R(n)log(R)) many random samples (taken independently uniformly distributed within C-R) yields a sampling set for V-R,(delta)(Phi) with high probability. We give explicit estimates of all involved constants in terms of the generators phi(1),...,phi. (C) 2018 Elsevier Inc. All rights reserved.