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A note on Gabor orthonormal bases

Author: Yun-Zhang Li
Journal: Proc. Amer. Math. Soc. 133 (2005), 2419-2428
MSC (2000): Primary 42C40
Published electronically: February 25, 2005
MathSciNet review: 2138885
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Abstract: 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 $\vert\mbox{supp}(g)\vert=1$.

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Yun-Zhang Li
Affiliation: School of Applied Mathematics and Physics, Beijing University of Technology, Beijing, 100022, People’s Republic of China

Keywords: Gabor orthonormal basis, tiling
Received by editor(s): December 3, 2003
Received by editor(s) in revised form: April 19, 2004
Published electronically: February 25, 2005
Additional Notes: This research was supported by the National Natural Science Foundation of China, and the Natural Science Foundation of Beijing
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.