On Kadec' \(1/4\)-theorem and the stability of Gabor frames
The well known Kadets' 1/4-theorem shows that if \(\{\lambda_n\}\) is a sequence of real numbers for which \(|\lambda_n- n|\leq L<{1\over 4}\), \(n\in\mathbb{Z}\), then \(\{e^{i\lambda_n \omega}\}\) forms a Riesz basis for \(L^2[-\pi,\pi]\). \textit{S. Favier} and \textit{R. Zalik} [ibid. 2, No. 2, 160-173 (1995; Zbl 0829.46006)] gave some multivariate version of Kadets' 1/4-theorem. In the present paper the authors improve this result: Let \(k= (k_1,k_2,\dots, k_d)\in \mathbb{Z}^d\), \(\lambda_k= (\lambda_{k_1},\lambda_{k_2},\dots, \lambda_{k_d})\in \mathbb{R}^d\). If \(\{\lambda_k\}\) is a sequence in \(\mathbb{R}^d\) such that \(|k_s- \lambda_{k_s}|\leq L<{1\over 4}\) for \(s= 1,2,\dots, d\), \(k\in \mathbb{Z}^d\), then \(\{e^{i\langle\lambda_k, \omega\rangle}: k\in \mathbb{Z}^d\}\) forms a Riesz basis for \(L^2[-\pi, \pi]^d\) with frame bounds \((2\pi)^d[1- B_1(L)]^{2d}\) and \((2\pi)^d[1+ B_1(L)]^{2d}\), where \(B_1(L)= 1-\cos(\pi L)+ \sin(\pi L)\). The authors also study the stability of Gabor frames.
scientific article; zbMATH DE number 1382919