Reconstruction from convolution random sampling in local shift invariant spaces

Abstract

In this paper, we consider the problem of reconstructing functions in local multiply generated shift invariant spaces from convolution random samples. The sampling set is randomly chosen with one kind of probability distribution over a bounded cube and the available data are the sampled convolution of the original function on the sampling set. We obtain an explicit reconstruction formula. This reconstruction formula succeeds with overwhelming probability when the sampling size is sufficiently large.