Random sampling and reconstruction in multiply generated shift-invariant spaces

Shift-invariant spaces play an important role in approximation theory, wavelet analysis, finite elements, etc. In this paper, we consider the stability and reconstruction algorithm of random sampling in multiply generated shift-invariant spaces V-p(Phi). Under some decay conditions of the generator Phi, we approximate V-p(Phi) with finite-dimensional subspaces and prove that with overwhelming probability, the stability of sampling set conditions holds uniformly for all functions in certain compact subsets of V-p(Phi) when the sampling size is sufficiently large. Moreover, we show that this stability problem is connected with properties of the random matrix generated by Phi. In the end, we give a reconstruction algorithm for the random sampling of functions in V-p(Phi).