Local reconstruction for sampling in shift-invariant spaces

The local reconstruction from samples is one of most desirable properties for many applications in signal processing, but it has not been given as much attention. In this paper, we will consider the local reconstruction problem for signals in a shift-invariant space. In particular, we consider finding sampling sets X such that signals in a shift-invariant space can be locally reconstructed from their samples on X. For a locally finite-dimensional shift-invariant space V we show that signals in V can be locally reconstructed from its samples on any sampling set with sufficiently large density. For a shift-invariant space V(ϕ1, ..., ϕN) generated by finitely many compactly supported functions ϕ1, ..., ϕN, we characterize all periodic nonuniform sampling sets X such that signals in that shift-invariant space V(ϕ1, ..., ϕN) can be locally reconstructed from the samples taken from X. For a refinable shift-invariant space V(ϕ) generated by a compactly supported refinable function ϕ, we prove that for almost all $(x_0, x_1)\in [0,1]^2$, any signal in V(ϕ) can be locally reconstructed from its samples from $\{x_0, x_1\}+{\mathbb Z}$ with oversampling rate 2. The proofs of our results on the local sampling and reconstruction in the refinable shift-invariant space V(ϕ) depend heavily on the linear independent shifts of a refinable function on measurable sets with positive Lebesgue measure and the almost ripplet property for a refinable function, which are new and interesting by themselves.