Kadec-1/4 Theorem for Sinc Bases

TL;DR

This paper establishes conditions under which sinc-based systems form Riesz bases for Paley-Wiener spaces, extending classical results with new parameter bounds and stability criteria.

Contribution

It introduces novel bounds for sinc systems to be Riesz bases, considering perturbations and specific decay rates of the sampling points.

Findings

Sinc systems form Riesz bases under certain decay conditions.

Stability of sinc bases is maintained with bounded perturbations.

New bounds depend on the Lamb-Oseen constant and decay rate.

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.