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 ${\left\{\mathrm{sinc}({\lambda }_n-t)\right\}}_{n\in \mathbb{N}}$ is a Riesz basis for $P{W}_{\pi }$. With the second result, we study the stability of ${\left\{\mathrm{sinc}({\lambda }_n-t)\right\}}_{n\in \mathbb{Z}}$ 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 $\{\mathrm{sinc}({\lambda }_n-t){\}}_{n\in \mathbb{Z}}$ forms a Riesz basis for $P{W}_{\pi }$. Here $\alpha$ is the Lamb-Oseen constant.