Gabor orthonormal bases generated by the unit cubes

Abstract

We consider the problem in determining the countable sets $\Lambda$ in the time-frequency plane such that the Gabor system generated by the time-frequency shifts of the window $\chi_{[0,1]^d}$ associated with $\Lambda$ forms a Gabor orthonormal basis for $ L^2({\Bbb R}^d)$. We show that, if this is the case, the translates by elements $\Lambda$ of the unit cube in ${\Bbb R}^{2d}$ must tile the time-frequency space ${\Bbb R}^{2d}$. By studying the possible structure of such tiling sets, we completely classify all such admissible sets $\Lambda$ of time-frequency shifts when $d=1,2$. Moreover, an inductive procedure for constructing such sets $\Lambda$ in dimension $d\ge 3$ is also given. An interesting and surprising consequence of our results is the existence, for $d\geq 2$, of discrete sets $\Lambda$ with ${\mathcal G}(\chi_{[0,1]^d},\Lambda)$ forming a Gabor orthonormal basis but with the associated "time"-translates of the window $\chi_{[0,1]^d}$ having significant overlaps.