Separation of zeros and a Hermite interpolation based frame algorithm for band limited functions

Abstract

It is shown that if a non-zero function $f\in B_{\sigma }$ has infinitely many double zeros on the real axis, then there exists at least one pair of consecutive zeros whose distance apart is greater than $\frac{\pi }{\sigma }{\tau }^{1/4}$, $\tau \approx 5.0625$. A frame algorithm is provided for reconstructing a function $f\in B_{\sigma }$ from its nonuniform samples $\{f^{(j)}(x_i):j=0,1,\dots ,k-1,i\in \mathbb{Z}\}$ with maximum gap condition, $\underset{i}{\sup }(x_{i+1}-x_i)=\delta <\frac{1}{\sigma }{c}_k^{1/2k}$, where $c_k$ is a Wirtinger-Sobolev constant, using Hermite interpolation.

12 pages