Kadec-1/4 Theorem for Sinc Bases

Abstract

In this paper we show two results. In the first result we consider $\lambda_n-n=\frac{A}{n^\alpha}$ for $n\in\mathbb N$; if $\alpha>1/2$ and $0<A<\frac{1}{\pi\sqrt{2 \sqrt{2}\zeta(2\alpha)}}$, the system $\left\{\operatorname{sinc}( \lambda_n - t)\right\}_{n\in\mathbb N}$ is a Riesz basis for $PW_{\pi}$. With the second result, we study the stability of $\left\{\operatorname{sinc}( \lambda_n - t)\right\}_{n\in\mathbb Z}$ for $\lambda_n\in\mathbb C$; if $|\lambda_n-n|\leqq L<\frac{1}{\pi}\, \sqrt\frac{3\alpha}{8}$, for all $n\in\mathbb Z$, then $\{\operatorname{sinc}(\lambda_n-t)\}_{n\in\mathbb Z}$ forms a Riesz basis for $PW_{\pi}$. Here $\alpha$ is the Lamb-Oseen constant.