Inequalities for irregular Gabor frames

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.