Gabor orthogonal bases and convexity

Abstract

Let $g(x)={\chi }_B(x)$ be the indicator function of a bounded convex set in ${\mathbb{R}}^d$, $d\ge 2$, with a smooth boundary and everywhere non-vanishing Gaussian curvature. Using a combinatorial appraoch we prove that if $d\ne 1\mathrm{mod}4$, then there does not exist $S\subset {\mathbb{R}}^{2d}$ such that ${\{g(x-a)e^{2\pi ix\cdot b}\}}_{(a,b)\in S}$ is an orthonormal basis for $L^2({\mathbb{R}}^d)$.

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• Non-separable lattices, Gabor orthonormal bases and Tilings