SHAPIRO TYPE INEQUALITIES FOR THE WEINSTEIN AND THE WEINSTEIN-GABOR TRANSFORMS

Authors : NEJIB BEN SALEM, Amgad RASHED NASR

Pages : 68-76

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Publication Date : 2017-04-01

Article Type : Research Paper

Abstract :The aim of this paper is to prove new uncertainty principles for the Weinstein and the Weinstein-Gabor transforms associated with the Weinstein operator dened on the half  space $\mathbb{R}^d_{+}$ by $\Delta_W =\sum_{i=1}^{d } \frac{\partial}{\partial x_i^2}+ \frac{2\alpha+1}{x_{d}}\frac{\partial}{\partial x_{d-1}};\ \ \ \ \ d\ge2,\ \alpha>-1/2.$ More precisely, we give a Shapiro-type uncertainty inequality for the Weinstein transform that is, for $s>0$ and $\{\phi_n\}_n$ be an orthonormal sequence in $L^2_\alphainsert ignore into journalissuearticles values(\mathbb{R}^d_{+});$, $\sum_{n=1}^Ninsert ignore into journalissuearticles values(\Vert \vert x\vert^s \phi_n\Vert_{{L_\alpha^2insert ignore into journalissuearticles values(\mathbb{R}^d_{+});}}^{2}+ \Vert \vert\xi\vert^s \mathcal{F}_Winsert ignore into journalissuearticles values(\phi_n);\Vert_{{L_\alpha^2insert ignore into journalissuearticles values(\mathbb{R}^d_{+});}}^{2 });\geq KN^{1+\frac{s}{2\alpha+d+1}},$ where $K$ is a constant which depends only on $d$; $s$ and $\alpha$. Next, we establish an analogous inequality for the Weinstein-Gabor transform
Keywords : Weinstein operator, Shapiro uncertainty inequality, time frequency concentration