Gabor orthogonal bases and convexity

TL;DR

This paper proves that for certain convex sets with smooth boundaries and non-zero Gaussian curvature, Gabor orthonormal bases cannot exist in dimensions where d is not congruent to 1 mod 4, using a combinatorial approach.

Contribution

It establishes a non-existence result for Gabor orthonormal bases associated with indicator functions of convex sets in specific dimensions, extending understanding of basis constructions.

Findings

No Gabor orthonormal bases for convex sets when d ≠ 1 mod 4

Uses combinatorial methods to prove non-existence

Applicable to convex sets with smooth boundary and non-vanishing curvature

Abstract

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