Gabor orthonormal bases generated by the unit cubes
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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.