A new sampling density condition for shift-invariant spaces

Abstract

Let $X=\{x_i:i\in \mathbb{Z}\}$, $\cdots <x_{i-1}<x_i<x_{i+1}<\dots$, be a sampling set which is separated by a constant $\gamma >0$. Under certain conditions on $\varphi$, it is proved that if there exists a positive integer $\nu$ such that $${\delta }_{\nu }:=\underset{i\in \mathbb{Z}}{\sup }(x_{i+\nu }-x_i)<\frac{\nu }{2\pi }{\left(\frac{c_k^2}{M_{2k}}\right)}^{\frac{1}{4k}},$$ then every function belonging to a shift-invariant space $V(\varphi )$ can be reconstructed stably from its nonuniform sample values $\{f^{(j)}(x_i):j=0,1,\dots ,k-1,i\in \mathbb{Z}\}$, where $c_k$ is a Wirtinger-Sobolev constant and $M_{2k}$ is a constant in Bernstein-type inequality of $V(\varphi )$. Further, when $k=1$, the maximum gap ${\delta }_{\nu }<\nu$ is sharp for certain shift-invariant spaces.

20 pages