Irregular Sampling on Shift Invariant Spaces

Abstract

Let V(φ) be a shift invariant subspace of $L^{2}(\mathbb{R})$ with a Riesz or frame generator φ(t). We take φ(t) suitably so that the regular sampling expansion : $f(t) = \sum _{n\in \mathbb{Z}}f(n)S(t-n)$ holds on V(φ). We then find conditions on the generator φ(t) and various bounds of the perturbation $\{ \delta _n \}_{n \in \mathbb{Z}}$ under which an irregular sampling expansion: ƒ(t) = $\sum_{n \in \mathbb{Z}} f(n+ \delta_n)S_n(t)$ holds on V(φ). Some illustrating examples are also provided.