A note on Gabor orthonormal bases

The study of Gabor bases of the form $\{\, e^{-2\pi i\langle\lambda,\,\cdot\rangle}g(\cdot-m):\,\lambda,\, m\in {\mathbb Z}^n\,\}$ for $L^2({\mathbb R}^n)$ has interested many mathematicians in recent years. Alex Losevich and Steen Pedersen in 1998, Jeffery C. Lagarias, James A. Reeds and Yang Wang in 2000 independently proved that, for any fixed positive integer $n$, $\{\, e^{-2\pi i\langle\lambda,\,\cdot\rangle}:\,\lambda\in \Lambda\,\}$ is an orthonormal basis for $L^2([0,\,1]^n)$ if and only if $\{\, [0,\,1]^n+\lambda:\, \lambda\in \Lambda\,\}$ is a tiling of ${\mathbb R}^n$. Palle E. T. Jorgensen and Steen Pedersen in 1999 gave an explicit characterization of such $\Lambda$ for $n=1$, $2$, $3$. Inspired by their work, this paper addresses Gabor orthonormal bases of the form $\{\, e^{-2\pi i\langle\lambda,\,\cdot\rangle}g(\cdot-m):\,\lambda\in\Lambda,\, m\in {\mathbb Z}^n\,\}$ for $L^2({\mathbb R}^n)$ and some other related problems, where $\Lambda$ is as above. For a fixed $n\in \{\, 1,\, 2,\, 3\,\}$, the generating function $g$ of a Gabor orthonormal basis for $L^2({\mathbb R}^n)$ corresponding to the above $\Lambda$ is characterized explicitly provided that ${\mbox{supp}}(g)=[a_1,\,b_1]\times\cdots\times [a_n,\, b_n]$, which is new even if $\Lambda={\mathbb Z}^n$; a Shannon type sampling theorem about such $\Lambda$ is derived when $n=2$, $3$; for an arbitrary positive integer $n$, an explicit expression of the $g$with $\{\, e^{-2\pi i\langle\lambda,\,\cdot\rangle}g(\cdot-m):\,\lambda,\, m\in {\mathbb Z}^n\,\}$ being an orthonormal basis for $L^2({\mathbb R}^n)$ is obtained under the condition that $\vert\mbox{supp}(g)\vert=1$.