Completeness of the Grid Translates of Functions |
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Authors: | Swagato K Ray Rudra P Sarkar Yitzhak Weit |
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Institution: | 1. Stat-Math Unit, Indian Statistical Institute, Calcutta, 700108, India 2. Department of Mathematics, University of Haifa, Haifa, Israel
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Abstract: | If $f\in L^{p}(\mathbb{R}^{d})$ is a bounded real valued continuous function which has a unique maximum or a unique minimum at a point $x_{0}\in \mathbb{R}^{d}$ and if the inverse image of the neighborhoods of f(x 0) shrinks regularly to x 0, then $\mathrm{ span }\{f^{m}(x-2^{-m}\varSigma_{i=1}^{d} j_{i} e_{i})\mid m\in\mathbb{N}, j_{i}\in\mathbb{Z}\}$ is a dense subset of $L^{p}(\mathbb{R}^{d}), 1\le p<\infty$ where f m (x)=f(x) m and {e i } is the natural basis of $\mathbb{R}^{d}$ . The result extends to all homogeneous groups, Riemannian symmetric spaces of noncompact type, Damek-Ricci spaces etc. |
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