PlanetMath: Gabor frame

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Gabor frame

(Definition)

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 is an invertible matrix, and let $\pi(\xi,\phi)f=e^{2 \pi i \xi x}f(x-\phi)$

Definition 1 Let $g \in L^2(\mathbb{R}^d)$ be a nonzero window, and let $\lambda \in \Lambda$ , then

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

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

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

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

$\displaystyle \pi( \xi, \phi ) \boldsymbol{g}=e^{2 \pi \i x \cdot \xi}\left( g_1(x-\phi),g_2(x-\phi),...,g_n(x-\phi) \right)$

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

$\displaystyle \left\langle \boldsymbol{f},\boldsymbol{h} \right\rangle_{L^2(\ma... ...C}^n)} = \sum_{i=1}^n \left\langle f_i , h_i \right\rangle_{L^2(\mathbb{R}^d)}$

Bibliography

1: Karlheinz Gröchenig, "Foundations of Time-Frequency Analysis," Birkhhäuser (2000)

"Gabor frame" is owned by ErlendA.

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Also defines:

Gabor frame, Gabor super-frame, Vector-valued Gabor frame

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Cross-references: inequalities, functions, matrix, invertible, lattice, frame, theory

This is version 2 of Gabor frame, born on 2007-05-23, modified 2007-05-23.
Object id is 9448, canonical name is GaborFrame.
Accessed 882 times total.

Classification:

AMS MSC:

46C99 (Functional analysis :: Inner product spaces and their generalizations, Hilbert spaces :: Miscellaneous)

Pending Errata and Addenda

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