Wiener amalgam spaces for the fundamental identity of Gabor analysis
By Hans G. Feichtinger, Franz Luef. 2005
67 citations recorded by the source corpus.
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 Rie el?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.
Subfields: Applied Mathematics, Computer Vision and Pattern Recognition
View Paper: https://www.semanticscholar.org/paper/7988187a5a27e53ac668b0138aec84f29316cb9f
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In Applied Mathematics
- Karlheinz Gröchenig
- Hans G. Feichtinger
- Franz Luef
- Götz E. Pfander
- Elena Cordero
- Qiyu Sun
- J. Toft
- Christopher Heil
In Computer Vision and Pattern Recognition
- Karlheinz Gröchenig
- Hans G. Feichtinger
- Franz Luef
- Götz E. Pfander
- Elena Cordero
- Qiyu Sun
- Christopher Heil
- J. Toft
Citations in the corpus
- Banach Gelfand Triples for Gabor Analysis (2008, background): “They are, however, all consequences of the fundamental identity of Gabor analysis extensively studied in [ 19 ].”
- Projective modules over noncommutative tori are multi-window Gabor frames for modulation spaces (2008, background): “We have shown in [21] that if f ∈ L(2)(R) and gi ∈ M 1 v (R ) then (Vgif ·Vgif(λ))λ is absolutely convergent for i = 1, .”
- The Mystery of Gabor Frames (2014, methodology): “For more information we refer to the original literature [29,33,70].”
- Gabor Frames in Finite Dimensions (2013, background): “Taking inner products of the left hand and the right hand side of (16) with x̃∈CG shows that Janssen’s representation implies the so-called fundamental identity in Gabor analysis (FIGA) (18) below, see also [35, 45].”