On the stability of multivariate trigonometric systems
type: Item
instance of: scholarly article
title: On the stability of multivariate trigonometric systems
zbMATH Open document ID: 0930.42017
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-10-25T00:00:00Z
review text: Kadec's 1/4-theorem says that if \(\{\lambda_n: n \in \mathbb{Z}\}\) is a sequence of real numbers for which \(|\lambda_n - n|\leq L < 1/4\), then \(\{e^{i \lambda_n \omega}: n \in \mathbb{Z}\}\) forms a Riesz basis for \(L^2[-\pi, \pi]\). \textit{S. J. Favier} and the reviewer [Appl. Comput. Harmon. Anal. 2, No. 2, 160-173 (1995; Zbl 0829.46006)] and \textit{C. K. Chui} and \textit{X. Shi} [Appl. Comput. Harmon. Anal. 3, No. 3, 283-287 (1996; Zbl 0876.42019)] extended this result to the multivariate case and provided frame bounds, which are not given in Kadec's theorem. But these bounds are very large, and this implies that the perturbations \(\lambda_n -n\) that preserve the Riesz basis property must be very small. In the paper under review optimal frame bounds are found. An extension to Fourier frames is also given. The authors have also published the main result of this paper, with the same proof, as the main result of [Appl. Comput. Harmon. Anal. 7, 239-242 (1999), article ID acha.1998.0263].
Mathematics Subject Classification ID: 42C40
zbMATH DE Number: 1332882
zbMATH Keywords: Kadec's 1/4-theorem
zbMATH Keywords: stability bound
zbMATH Keywords: frame
zbMATH Keywords: Riesz basis
author: Wenchang Sun
author: Xing-Wei Zhou
reviewed by: Richard A. Zalik
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6386
OpenAlex ID: W1977762217
cites work: Stability theorems for Fourier frames and wavelet Riesz bases
cites work: A Paley-Wiener Theorem for Frames
cites work: Q4373881
cites work: On stability bounds of perturbed multivariate trigonometric systems
cites work: On the stability of frames and Riesz bases
cites work: Q5593353
cites work: Q3955832
DOI: 10.1006/JMAA.1999.6386
description: scientific article; zbMATH DE number 1332882