On a (No Longer) New Segal Algebra: A Review of the Feichtinger Algebra

Jakobsen, Mads Sielemann

doi:10.1007/s00041-018-9596-4

Cited by 79 publications

(87 citation statements)

References 108 publications

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“…The Feichtinger algebra has many different descriptions, see [27]. Functions in S 0 (G) are continuous.…”

Section: Time-frequency Analysis and Heisenberg Modulesmentioning

confidence: 99%

“…Denote by F the usual Fourier transform S 0 (G) → S 0 ( G). Since the Poisson summation formula holds for functions in S 0 (G) [27,Theorem 5.7 (iii)] and S 0 (G) is an algebra under pointwise multiplication, we have that the Poisson summation formula holds for the function t → ξ(tx)η(λ −1 tx), with x ∈ G, ω ∈ G. We do the following calculation, where the Poisson summation formula is applied in the fifth equality:…”

Section: Connecting Heisenberg Modules and Vector Bundlesmentioning

confidence: 99%

“…Feichtinger [15]. It has several characterizations [27], one of them being that it consists of all functions ξ ∈ L 2 (G) for which G× G | ξ, π(z)ξ | dz < ∞.…”

Section: Introductionmentioning

confidence: 99%

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The Balian–Low theorem for locally compact abelian groups and vector bundles

Enstad

2020

Journal de Mathématiques Pures et Appliquées

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Let Λ be a lattice in a second countable, locally compact abelian group G with annihilator Λ ⊥ ⊆ G. We investigate the validity of the following statement: For every η in the Feichtinger algebra S 0 (G), the Gabor system {Mτ T λ η} λ∈Λ,τ ∈Λ ⊥ is not a frame for L 2 (G). When G = R and Λ = αZ, this statement is a variant of the Balian-Low theorem. Extending a result of R. Balan, we show that whether the statement generalizes to (G, Λ) is equivalent to the nontriviality of a certain vector bundle over the compact space (G/Λ) × ( G/Λ ⊥ ). We prove this equivalence using a connection between Gabor frames and Heisenberg modules. More specifically, we show that the Zak transform can be viewed as an isomorphism of certain Hilbert C * -modules. As an application, we prove a new Balian-Low theorem for the group R × Qp, where Qp denotes the p-adic numbers.

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“…The Feichtinger algebra has many different descriptions, see [27]. Functions in S 0 (G) are continuous.…”

Section: Time-frequency Analysis and Heisenberg Modulesmentioning

confidence: 99%

Section: Connecting Heisenberg Modules and Vector Bundlesmentioning

confidence: 99%

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The Balian–Low theorem for locally compact abelian groups and vector bundles

Enstad

2020

Journal de Mathématiques Pures et Appliquées

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“…There are plenty of equivalent characterizations for the Feichtinger algebra, we refer the interested reader to cf. [35].…”

Section: Basic Properties Of Mmentioning

confidence: 99%

Linear Perturbations of the Wigner Transform and the Weyl Quantization

Bayer

Cordero

Gröchenig

et al. 2020

Applied and Numerical Harmonic Analysis

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We study a class of quadratic time-frequency representations that, roughly speaking, are obtained by linear perturbations of the Wigner transform. They satisfy Moyal's formula by default and share many other properties with the Wigner transform, but in general they do not belong to Cohen's class. We provide a characterization of the intersection of the two classes. To any such time-frequency representation, we associate a pseudodifferential calculus. We investigate the related quantization procedure, study the properties of the pseudodifferential operators, and compare the formalism with that of the Weyl calculus.The group of invertible, real-valued 2d × 2d matrices is denoted byand we denote the transpose of an inverse matrix byLet J denote the canonical symplectic matrix in R 2d , namelyObserve that, for z = (z 1 , z 2 ) ∈ R 2d , we have Jz = J (z 1 , z 2 ) = (z 2 , −z 1 ) , J −1 z = J −1 (z 1 , z 2 ) = (−z 2 , z 1 ) = −Jz, and J 2 = −I 2d×2d . Function spacesRecall that C 0 (R d ) denotes the class of continuous functions on R d vanishing at infinity. Modulation spaces. Fix a non-zero window g ∈ S(R d ) and 1 ≤ p, q ≤ ∞., (with obvious modifications for p = ∞ or q = ∞). If p = q, we write M p instead of M p, p .

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“…For any LCA group G the Feichtinger algebra S 0 (G) [7,15,22] (sometimes denoted by M 1 (G)) is the set of functions given by…”

Section: The Feichtinger Algebramentioning

confidence: 99%

Sampling and periodization of generators of Heisenberg modules

Jakobsen¹,

Luef

2019

Int. J. Math.

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This paper considers generators of Heisenberg modules in the case of twisted group C * -algebras of closed subgroups of locally compact abelian groups and how the restrction and/or periodization of these generators yield generators for other Heisenberg modules. Since generators of Heisenberg modules are exactly the generators of (multi-window) Gabor frames, our methods are going to be from Gabor analysis. In the latter setting the procedure of restriction and periodization of generators is well known. Our results extend this established part of Gabor analysis to the general setting of locally compact abelian groups. We give several concrete examples where we demonstrate some of the consequences of our results. Finally, we show that vector bundles over an irrational noncommutative torus may be approximated by vector bundles for finite-dimensional matrix algebras that converge to the irrational noncommutative torus with respect to the module norm of the generators, where the matrix algebras converge in the quantum Gromov-Hausdorff distance to the irrational noncommutative torus.

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On a (No Longer) New Segal Algebra: A Review of the Feichtinger Algebra

Cited by 79 publications

References 108 publications

The Balian–Low theorem for locally compact abelian groups and vector bundles

The Balian–Low theorem for locally compact abelian groups and vector bundles

Linear Perturbations of the Wigner Transform and the Weyl Quantization

Sampling and periodization of generators of Heisenberg modules

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