On the stability of multivariate trigonometric systems
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].
scientific article; zbMATH DE number 1332882