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Inequalities for irregular Gabor frames
Authors:Lili Zang and Wenchang Sun
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 {eimbxg(x-na): m,n ? \Bbb Z}\{e^{imbx}g(x-na): m,n\in{\Bbb Z}\} is a Gabor frame for L2(\Bbb R)L^2({\Bbb R}) with frame bounds A and B, then the following two inequalities hold: A £ \frac2pb?n ? \Bbb Z|g(x-na)|2B,     a.e.A\le \frac{2\pi}{b}\sum_{n\in{\Bbb Z}}\vert g(x-na)\vert^2\le B, \quad a.e. and A £ \frac1a?m ? \Bbb Z|^(g)](w-mb)|2B,     a.e.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 è1 £ kr{eiáx, l?gk(x-m): m ? Dk, l ? Lk }\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 Rd{\Bbb R}^d and gk ? L2(\Bbb Rd)g_k\in{L^2{(\Bbb R}^d)} , 1 ≤ kr.
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