Irregular sampling on shift invariant spaces

Abstract

For any $\phi(t)$ in $L^{2}(\Real)$, let $V(\phi)$ be the closed shift invariant subspaces of $L^{2}(\Real)$ spanned by integer translates $\{\phi(t-n):n\in \mathbb{Z} \}$ of $\phi(t)$. Assuming that the shift invariant space $V(\phi)$ is an RKHS and regular expansion is possible, we find conditions that irregular sampling expansion \begin{equation*} f(t) = \sum_{n\in \mathbb{Z}} f(n+\delta_n) S_n(t) \end{equation*} holds.