Sampling expansion in shift invariant spaces

Abstract

For any phi(t) in L-2(R), let V (phi) be the closed shift invariant subspace of L-2(R) spanned by integer translates {phi(t - n) : n is an element of Z} of phi(t). Assuming that phi(t) is a frame or a Riesz generator of V(phi), we first find conditions under which V(phi) becomes a reproducing kernel Hilbert space. We then find necessary and sufficient conditions under which an irregular or a regular shifted sampling expansion formula holds on V(phi) and obtain truncation error estimates of the sampling series. We also find a sufficient condition for a function in L-2(R) that belongs to a sampling subspace of L-2(R). Several illustrating examples are also provided.