## Kyoto Journal of Mathematics

### Gabor families in $l^{2}(\mathbb{Z}^{d})$

#### Abstract

This paper addresses Gabor families in $l^{2}(\mathbb{Z}^{d})$. The discrete Gabor families have interested many researchers due to their good potential for digital signal processing. Gabor analysis in $l^{2}(\mathbb{Z}^{d})$ is more complicated than that in $l^{2}(\mathbb{Z})$ since the geometry of the lattices generated by time-frequency translation matrices can be quite complex in this case. In this paper, we characterize window functions such that they correspond to complete Gabor families (Gabor frames) in $l^{2}(\mathbb{Z}^{d})$; obtain a necessary and sufficient condition on time-frequency translation for the existence of complete Gabor families (Gabor frames, Gabor Riesz bases) in $l^{2}(\mathbb{Z}^{d})$; characterize duals with Gabor structure for Gabor frames; derive an explicit expression of the canonical dual for a Gabor frame; and prove its norm minimality among all Gabor duals.

#### Article information

Source
Kyoto J. Math., Volume 52, Number 1 (2012), 179-204.

Dates
First available in Project Euclid: 19 February 2012

https://projecteuclid.org/euclid.kjm/1329684747

Digital Object Identifier
doi:10.1215/21562261-1503800

Mathematical Reviews number (MathSciNet)
MR2892772

Zentralblatt MATH identifier
1242.42026

Subjects
Primary: 42C15: General harmonic expansions, frames

#### Citation

Lian, Qiao-Fang; Li, Yun-Zhang. Gabor families in $l^{2}(\mathbb{Z}^{d})$. Kyoto J. Math. 52 (2012), no. 1, 179--204. doi:10.1215/21562261-1503800. https://projecteuclid.org/euclid.kjm/1329684747

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