Almost Diagonalization of $$\tau $$ τ -Pseudodifferential Operators with Symbols in Wiener Amalgam and Modulation Spaces

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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 .

“…These results have been put into the context of time-frequency analysis in [29] and have been generalized, see [15,30,31]. In this section we extend Theorem 19 with respect to MWDs.…”

Section: Theorem 19mentioning

confidence: 78%

Section: Theorem 8 ([1 Thm 151]) Letmentioning

confidence: 99%

See 1 more Smart Citation

Linear Perturbations of the Wigner Transform and the Weyl Quantization

Bayer

Cordero

Gröchenig

et al. 2020

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“…We already discussed in the Introduction how this general class of operators extends in a natural way the previous one, which can be recovered as the case τ = 1/2. We were able to obtain an identical result with an identical proof -apart from the substantial modifications in the preliminary lemmas -see [5] for the details. Λ ) is a Gabor frame for L 2 R d .…”

Section: Almost Diagonalization Of τ-Pseudodifferential Operatorsmentioning

confidence: 78%

“…In the subsequent Section 4 we report some results on almost diagonalization obtained by the author in a recent joint work with Elena Cordero and Fabio Nicola -see [5]. Mimicking the scheme which leads to define the Weyl transform, in [1] the authors consider a one-parameter family of time-frequency representations (τ-Wigner distributions) and also define the corresponding pseudodifferential operators Op τ via duality.…”

Section: Introductionmentioning

confidence: 99%

Almost Diagonalization of Pseudodifferential Operators

Trapasso

2019

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In this review we focus on the almost diagonalization of pseudodifferential operators and highlight the advantages that time-frequency techniques provide here. In particular, we retrace the steps of an insightful paper by Gröchenig, who succeeded in characterizing a class of symbols previously investigated by Seöstrand by noticing that Gabor frames almost diagonalize the corresponding Weyl operators. This approach also allows to give new and more natural proofs of related results such as boundedness of operators or algebra and Wiener properties of the symbol class. Then, we discuss some recent developments on the theme, namely an extension of these results to a more general family of pseudodifferential operators and similar outcomes for a symbol class closely related to Sjöstrand's one. This is a precopyedited version of a contribution published in Landscapes ofTime-Frequency Analysis (Boggiatto P. et al. (eds)) published by Birkhäuser, Cham. The definitive authenticated version is available online via https://doi.org/10.1007/978-3-030-05210-2 14.

Continuity Properties of Multilinear Localization Operators on Modulation Spaces

Teofanov

2019

We introduce multilinear localization operators in terms of the short-time Fourier transform, and multilinear Weyl pseudodifferential operators. We prove that such localization operators are in fact Weyl pseudodifferential operators whose symbols are given by the convolution between the symbol of the localization operator and the multilinear Wigner transform. For such interpretation we use the kenrel theorem for the Gelfand-Shilov space. Furthermore, we study the continuity properties of the multilinear localization operators on modulation spaces. Our results extend some known results when restricted to the linear case.1991 Mathematics Subject Classification. 47G30, 35S05, 46F05, 44A35.