A new sampling density condition for shift-invariant spaces

Let $X=\{x_i:i\in\mathbb{Z}\}$, $\dots<x_{i-1}<x_i<x_{i+1}<\dots$, be a sampling set which is separated by a constant $\gamma>0$. Under certain conditions on $\phi$, it is proved that if there exists a positive integer $\nu$ such that $$\delta_\nu:=\sup\limits_{i\in\mathbb{Z}}(x_{i+\nu}-x_i)<\dfrac{\nu}{2\pi}\left(\dfrac{c_{k}^2}{M_{2k}}\right)^{\frac{1}{4k}},$$ then every function belonging to a shift-invariant space $V(\phi)$ 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 $V(\phi)$. Further, when $k=1$, the maximum gap $\delta_\nu<\nu$ is sharp for certain shift-invariant spaces.