In the last decade it has become clear that one of the central themes within Gabor analysis (with respect to general time-frequency lattices) is a duality theory for Gabor frames, including the Wexler-Raz biorthogonality condition, the Ron-Shen duality principle and the Janssen representation of a Gabor frame operator. All these results are closely connected with the so-called Fundamental Identity of Gabor Analysis, which we derive from an application of Poisson's summation formula for the symplectic Fourier transform. The new aspect of this presentation is the description of the range of validity of this Fundamental Identity of Gabor Analysis using Wiener amalgam spaces and Feichtinger's algebra S0(Rd). Our approach is inspired by Rieffel's use of the Fundamental Identity of Gabor Analysis in the study of operator algebras generated by time-frequency shifts along a lattice, which was later independently rediscovered by Tolmieri/Orr, Janssen, and Daubechies et al., and Feichtinger/Kozek at various levels of generality, in the context of Gabor analysis.
@article{urn:eudml:doc:41784,
title = {Wiener amalgam spaces for the fundamental identity of Gabor analysis.},
journal = {Collectanea Mathematica},
volume = {57},
year = {2006},
pages = {233-253},
zbl = {1135.39303},
mrnumber = {MR2264211},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41784}
}
Feichtinger, Hans G.; Luef, Franz. Wiener amalgam spaces for the fundamental identity of Gabor analysis.. Collectanea Mathematica, Tome 57 (2006) pp. 233-253. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41784/