Abstract.
In [C.K. Chui and X.L. Shi, Inequalities of Littlewood-Paley type for frames and wavelets, SIAM J. Math. Anal., 24 (1993), 263–277], the authors proved that if \(\{e^{imbx}g(x-na): m,n\in{\Bbb Z}\}\) is a Gabor frame for \(L^2({\Bbb R})\) with frame bounds A and B, then the following two inequalities hold: \(A\le \frac{2\pi}{b}\sum_{n\in{\Bbb Z}}\vert g(x-na)\vert^2\le B, \quad a.e.\) and \(A\le \frac{1}{a}\sum_{m\in{\Bbb Z}}\vert \hat{g}(\omega-mb)\vert^2\le B, \quad a.e.\). In this paper, we show that similar inequalities hold for multi-generated irregular Gabor frames of the form \(\bigcup_{1\le k\le r}\{e^{i\langle x, \lambda\rangle}g_{k}(x-\mu):\, \mu\in \Delta_k, \lambda\in\Lambda_k \}\), where Δ k and Λ k are arbitrary sequences of points in \({\Bbb R}^d\) and \(g_k\in{L^2{(\Bbb R}^d)}\), 1 ≤ k ≤ r.
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Authors’ address: Lili Zang and Wenchang Sun, Department of Mathematics and LPMC, Nankai University, Tianjin 300071, China
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Zang, L., Sun, W. Inequalities for irregular Gabor frames. Monatsh Math 154, 71–81 (2008). https://doi.org/10.1007/s00605-007-0506-0
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DOI: https://doi.org/10.1007/s00605-007-0506-0