Locally finite dimensional shift-invariant spaces in $\mathbf {R}^d$

Abstract

We prove that a locally finite dimensional shift-invariant linear space of distributions must be a linear subspace of some shift-invariant space generated by finitely many compactly supported distributions. If the locally finite dimensional shift-invariant space is a subspace of the Hölder continuous space $C^\alpha$ or the fractional Sobolev space $L^{p, \gamma }$, then the superspace can be chosen to be $C^\alpha$ or $L^{p, \gamma }$, respectively.