Time-Frequency Analysis of Sjöstrand’s Class

Gröchenig, Karlheinz

doi:10.4171/rmi/471

Cited by 161 publications

(225 citation statements)

References 29 publications

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“…Compare Gröchenig's use of Jaffard's Lemma [117] to establish the related polynomial version of the Structure Theorem, as well as the use of noncommutative Wiener's lemmas for lattice Gabor frames in [89, Section 13.3], [97]. We also refer to the new results in [92,99,172] for a deeper understanding of the role of Wiener's lemmas in time-frequency analysis.…”

Section: Structure Of the Canonical Dual Framementioning

confidence: 99%

History and Evolution of the Density Theorem for Gabor Frames

Heil

2007

J Fourier Anal Appl

173

137

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ABSTRACT. The Density Theorem for Gabor Frames is one of the fundamental results of time-frequency analysis. This expository survey attempts to reconstruct the long and very involved history of this theorem and to present its context and evolution, from the onedimensional rectangular lattice setting, to arbitrary lattices in higher dimensions, to irregular Gabor frames, and most recently beyond the setting of Gabor frames to abstract localized frames. Related fundamental principles in Gabor analysis are also surveyed, including the Wexler-Raz biorthogonality relations, the Duality Principle, the Balian-Low Theorem, the Walnut and Janssen representations, and the Homogeneous Approximation Property. An extended bibliography is included.

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Section: Structure Of the Canonical Dual Framementioning

confidence: 99%

History and Evolution of the Density Theorem for Gabor Frames

Heil

2007

J Fourier Anal Appl

173

137

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“…Analogous to [19] we define the modulation space M -norm, we obtain a class of symbols that leads to compact operators, see [3].…”

Section: Tools From Time-frequency Analysismentioning

confidence: 99%

“…(b) Proofs presented in a setting of LCA groups show in some sense "what is really going on". The derivations are stripped off of lengthy analytic estimates and replaced by a time-frequency (phase space) approach based on the ideas from [19]. Admittedly our approach requires more conceptual effort.…”

Section: Introductionmentioning

confidence: 99%

Pseudodifferential operators on locally compact abelian groups and Sjöstrand's symbol class

Gröchenig¹,

Strohmer²

2007

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We investigate pseudodifferential operators on arbitrary locally compact abelian groups. As symbol classes for the Kohn-Nirenberg calculus we introduce a version of Sjöstrand's class. Pseudodifferential operators with such symbols form a Banach algebra that is closed under inversion. Since "hard analysis" techniques are not available on locally compact abelian groups, a new time-frequency approach is used with the emphasis on modulation spaces, Gabor frames, and Banach algebras of matrices. Sjöstrand's original results are thus understood as a phenomenon of abstract harmonic analysis rather than "hard analysis" and are proved in their natural context and generality.

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“…The Gohberg-Baskakov-Sjöstrand class [3,19,20]. The inverse-closed property has important applications in dual wavelet frames, dual Gabor frames, and algebra of pseudo-differential operators [1,5,8,10,12,13,18,21]. The reader may refer to [6,11,16] and references therein for historical remarks, recent advances and applications.…”

Section: Introductionmentioning

confidence: 99%