# Gabor frame

One may be interested in Gabor frames and its related theory if one looks further into the frame framework. First, denote a lattice by $\Lambda=A\mathbb{Z}^{2d}$, where $A$ is an invertible matrix, and let $\pi(\xi,\phi)f=e^{2\pi i\xi x}f(x-\phi)$

###### Definition.

Let $g\in L^{2}(\mathbb{R}^{d})$ be a nonzero window, and let $\lambda\in\Lambda$, then

 $G(g,\lambda)=\left\{\pi(\lambda)g:\lambda\in\Lambda\right\}$

is a Gabor system. If $G(g,\lambda)$ is a frame, it’s called a Gabor frame for $L^{2}(\mathbb{R}^{d})$

Supose now that one wants to look at a more general framework, and work with functions in $L^{2}(\mathbb{R}^{d},\mathbb{C}^{n})$. Then the definition above generalises to

###### Definition.

Let $\boldsymbol{g}\in L^{2}(\mathbb{R}^{d},\mathbb{C}^{n})$ be a nonzero window and let $\lambda\in\Lambda$, then

 $\boldsymbol{G}(\boldsymbol{g},\lambda)=\left\{\pi(\lambda)\boldsymbol{g}:% \lambda\in\Lambda\right\}$

is a Gabor super-frame if the frame inequalities hold, where

 $\pi(\xi,\phi)\boldsymbol{g}=e^{2\pi\i x\cdot\xi}\left(g_{1}(x-\phi),g_{2}(x-% \phi),...,g_{n}(x-\phi)\right)$

and for $\boldsymbol{f},\boldsymbol{h}\in L^{2}(\mathbb{R}^{d},\mathbb{C}^{n})$

 $\left\langle\boldsymbol{f},\boldsymbol{h}\right\rangle_{L^{2}(\mathbb{R}^{d},% \mathbb{C}^{n})}=\sum_{i=1}^{n}\left\langle f_{i},h_{i}\right\rangle_{L^{2}(% \mathbb{R}^{d})}$

## References

• 1 Karlheinz GrÃÂ¶chenig, ”Foundations of Time-Frequency Analysis,” BirkhhÃÂ¤user (2000)
Title Gabor frame GaborFrame 2013-03-22 17:08:28 2013-03-22 17:08:28 ErlendA (6587) ErlendA (6587) 5 ErlendA (6587) Definition msc 46C99 Gabor frame Gabor super-frame Vector-valued Gabor frame