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Time-frequency analysis of pseudodifferential operators

MONATSHEFTE FUR MATHEMATIK 2001年第2期133卷 143-156页

作者： Labate, D Georgia Inst Technol Sch Math Atlanta GA 30332 USA

In this paper we apply a time-frequency approach to the study of pseudodifferential operators. Both the Weyl and the Kohn-Nirenberg correspondences are considered. In order to quantify the time-frequency content of a function or distribution, we use certain function spaces called modulation spaces. We deduce a time-frequency characterization of the twisted product sigma#tau of two symbols sigma and tau, and we show that modulation spaces provide the natural setting to exactly control the time-frequency content of sigma#tau from the time-frequency content of sigma and tau. As a consequence, we discuss some boundedness and spectral properties of the corresponding operator with symbol sigma#tau.

关键词： Kohn-Niremberg correspondence modulation spaces pseudodifferential operators time-frequency analysis trace-class operators twisted product Weyl correspondence

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Quasi-Banach algebras and Wiener properties for pseudodifferential and generalized metaplectic operators

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JOURNAL OF PSEUDO-DIFFERENTIAL OPERATORS AND APPLICATIONS 2023年第1期14卷 1-26页

作者： Cordero, Elena Giacchi, Gianluca Univ Torino Dipartimento Matemat Via Carlo Alberto 10 I-10123 Turin Italy Univ Bologna Dipartimento Matemat Piazza Porta San Donato 5 I-40126 Bologna Italy Lausanne Univ Hosp Dept Diagnost & Intervent Radiol Rue Bugnon 46 CH-1011 Lausanne Switzerland Univ Lausanne Rue Bugnon 46 CH-1011 Lausanne Switzerland HES SO Valais Wallis Inst Syst Engn Sch Engn Rue Ind 21 CH-1950 Sion Switzerland Sense Innovat & Res Ctr Ave Provence 82 CH-1007 Lausanne Switzerland Sense Innovat & Res Ctr Ch Agasse 5 CH-1950 Sion Switzerland

We generalize the results for Banach algebras of pseudodifferential operators obtained by Grochenig and Rzeszotnik (Ann Inst Fourier 58:2279-2314, 2008) to quasi-algebras of Fourier integral operators. Namely, we introduce quasi-Banach algebras of symbol classes for Fourier integral operators that we call generalized metaplectic operators, including pseudodifferential operators. This terminology stems from the pioneering work on Wiener algebras of Fourier integral operators (Cordero et al. in J Math Pures Appl 99:219-233, 2013), which we generalize to our framework. This theory finds applications in the study of evolution equations such as the Cauchy problem for the Schrodinger equation with bounded perturbations, cf. (Cordero, Giacchi and Rodino in Wigner analysis of operators. Part II: Schrodinger equations, ).

关键词： Quasi-Banach algebras Pseudodifferential operators Metaplectic operators modulation spaces Short-time Fourier transform Wiener algebra

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Sampling of Operators

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JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS 2013年第3期19卷 612-650页

作者： Pfander, Goetz E. Jacobs Univ Bremen Sch Engn & Sci D-28759 Bremen Germany

Sampling and reconstruction of functions is a fundamental tool in science. We develop an analogous sampling theory for operators whose Kohn-Nirenberg symbols are bandlimited. We prove sampling theorems for in this sense bandlimited operators and show that our results generalize both, the classical sampling theorem, and the fact that a time-invariant operator is fully determined by its impulse response.

关键词： Generalized sampling Operator identification Channel measurement Paley-Wiener spaces modulation spaces

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Time-frequency analysis of Born-Jordan pseudodifferential operators

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JOURNAL OF FUNCTIONAL ANALYSIS 2017年第2期272卷 577-598页

作者： Cordero, Elena de Gosson, Maurice Nicola, Fabio Univ Turin Dipartimento Matemat Via Carlo Alberto 10 I-10123 Turin Italy Univ Vienna Fac Math Oskar Morgenstern Pl 1 A-1090 Vienna Austria Politecn Torino Dipartimento Sci Matemat Corso Duca Abruzzi 24 I-10129 Turin Italy

Born-Jordan operators are a class of pseudodifferential operators arising as a generalization of the quantization rule for polynomials on the phase space introduced by Born and Jordan in 1925. The weak definition of such operators involves the Born-Jordan distribution, first introduced by Cohen in 1966 as a member of the Cohen class. We perform a time frequency analysis of the Cohen kernel of the Born-Jordan distribution, using modulation and Wiener amalgam spaces. We then provide sufficient and necessary conditions for Born Jordan operators to be bounded on modulation spaces. We use modulation spaces as appropriate symbols classes. (C) 2016 Elsevier Inc. All rights reserved.

关键词： Time-frequency analysis Wigner distribution Born-Jordan distribution modulation spaces

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An extension of localization operators

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JOURNAL OF PSEUDO-DIFFERENTIAL OPERATORS AND APPLICATIONS 2024年第1期15卷 12-12页

作者： Boggiatto, Paolo Garello, Gianluca Univ Torino Dept Math Via Carlo Alberto 10 I-10123 Turin Italy

We review at first the role of localization operators as a meeting point of three different areas of research, namely: signal analysis, quantization and pseudo-differential operators. We extend then the correspondence between symbol and operator which characterizes localization operators to a more general situation, introducing the class of bilocalization operators. We show that this enlargement yields a quantization rule that is closed under composition. Some boundedness results are deduced both for localization and bilocalization operators. In particular for bilocalization operators we prove that square integrable symbols yield bounded operators on L(2 )and that the class of bilocalization operators with integrable symbols is a subalgebra of bounded operators on every fixed modulation space.

关键词： Localization operators Operators composition L-p Lebesgue spaces modulation spaces

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Symplectic analysis of time-frequency spaces

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JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES 2023年第1期177卷 154-177页

作者： Cordero, Elena Giacchi, Gianluca Univ Torino Dipartimento Matemat Via Carlo Alberto 10 I-10123 Turin Italy Univ Bologna Dipartimento Matemat Piazza Porta San Donato 5 I-40126 Bologna Italy HES SO Valais Wallis Inst Syst Engn Sch Engn Rue Ind 21 CH-1950 Sion Switzerland Lausanne Univ Hosp Rue Bugnon 46 CH-1011 Lausanne Switzerland Univ Lausanne Dept Diagnost & Intervent Radiol Rue Bugnon 46 CH-1011 Lausanne Switzerland Sense Innovat & Res Ctr Ave Provence 82 CH-1007 Lausanne Switzerland

We present a different symplectic point of view in the definition of weighted modulation spaces Mmp,q (Rd) and weighted Wiener amalgam spaces W (FLpm1, Lqm2 )(Rd). All the classical time-frequency representations, such as the short-time Fourier transform (STFT), the tau-Wigner distributions and the ambiguity function, can be written as metaplectic Wigner distributions mu(A)(f circle times g over bar ), where mu(A) is the metaplectic operator and A is the associated symplectic matrix. Namely, time-frequency representations can be represented as images of metaplectic operators, which become the real protagonists of time-frequency analysis. In [13], the authors suggest that any metaplectic Wigner distribution that satisfies the so-called shift-invertibility condition can replace the STFT in the definition of modulation spaces. In this work, we prove that shift-invertibility alone is not sufficient, but it has to be complemented by an upper-triangularity condition for this characterization to hold, whereas a lower-triangularity property comes into play for Wiener amalgam spaces. The shiftinvertibility property is necessary: Rihaczek and conjugate Rihaczek distributions are not shift-invertible and they fail the characterization of the above spaces. We also exhibit examples of shift-invertible distributions without upper-triangularity condition which do not define modulation spaces. Finally, we provide new families of time-frequency representations that characterize modulation spaces, with the purpose of replacing the time-frequency shifts with other atoms that allow to decompose signals differently, with possible new outcomes in applications.

关键词： Time-frequency analysis modulation spaces Wiener amalgam spaces Time-frequency representations Metaplectic group Symplectic group

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Decay and smoothness for eigenfunctions of localization operators

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2020年第2期492卷 124480-124480页

作者： Bastianoni, Federico Cordero, Elena Nicola, Fabio Politecn Torino Dipartimento Sci Matemat Corso Duca Abruzzi 24 I-10129 Turin Italy Univ Torino Dipartimento Matemat Via Carlo Alberto 10 I-10123 Turin Italy

We study decay and smoothness properties for eigenfunctions of compact localization operators A(a)(phi 1,phi 2). Operators A(a)(phi 1,phi 2) with symbols a in the wide modulation space M-p,M-infinity (containing the Lebesgue space L-p), p0 (subspaces of M-p,M-infinity(R(2)d), p>2d/s) the L-2-eigenfunctions of A(a)(phi 1,phi 2) are actually Schwartz functions. An important role is played by quasi-Banach Wiener amalgam and modulation spaces. As a tool, new convolution relations for modulation spaces and multiplication relations for Wiener amalgam spaces in the quasi-Banach setting are exhibited. (C) 2020 Elsevier Inc. All rights reserved.

关键词： Time-frequency analysis Localization operators Short-time Fourier transform Quasi-Banach spaces modulation spaces Wiener amalgam spaces

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Coorbit spaces and Banach frames on homogeneous spaces with applications to the sphere

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ADVANCES IN COMPUTATIONAL MATHEMATICS 2004年第1-2期21卷 147-180页

作者： Dahlke, S Steidl, G Teschke, G Univ Marburg Fachbereich Math & Informat D-35032 Marburg Germany Univ Mannheim Fak Math & Informat D-68131 Mannheim Germany Univ Bremen Fachbereich 3 D-28334 Bremen Germany

This paper is concerned with the construction of generalized Banach frames on homogeneous spaces. The major tool is a unitary group representation which is square integrable modulo a certain subgroup. By means of this representation, generalized coorbit spaces can be defined. Moreover, we can construct a specific reproducing kernel which, after a judicious discretization, gives rise to atomic decompositions for these coorbit spaces. Furthermore, we show that under certain additional conditions our discretization method generates Banach frames. We also discuss nonlinear approximation schemes based on the atomic decomposition. As a classical example, we apply our construction to the problem of analyzing and approximating functions on the spheres.

关键词： square integrable group representations time-frequency analysis atomic decompositions frames homogeneous spaces coorbit spaces modulation spaces nonlinear approximation spheres

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Algebras of symbols associated with the Weyl calculus for Lie group representations

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MONATSHEFTE FUR MATHEMATIK 2012年第1期167卷 13-33页

作者： Beltita, Ingrid Beltita, Daniel Romanian Acad Sim Stoilow Inst Math Res Unit 1 Bucharest Romania

We develop our earlier approach to the Weyl calculus for representations of infinite-dimensional Lie groups by establishing continuity properties of the Moyal product for symbols belonging to various modulation spaces. For instance, we prove that the modulation space of symbols M (a,1) is an associative Banach algebra and the corresponding operators are bounded. We then apply the abstract results to two classes of representations, namely the unitary irreducible representations of nilpotent Lie groups, and the natural representations of the semidirect product groups that govern the magnetic Weyl calculus. The classical Weyl-Hormander calculus is obtained for the Schrodinger representations of the finite-dimensional Heisenberg groups, and in this case we recover the results obtained by J. Sjostrand (Math Res Lett 1(2):185-192, 1994).

关键词： Weyl calculus Involutive Banach algebra Wiener property Lie group modulation spaces

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NLS in the modulation Space M2,q(R)

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JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS 2019年第4期25卷 1447-1486页

作者： Pattakos, N. Karlsruhe Inst Technol Inst Anal Dept Math D-76128 Karlsruhe Germany

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schrodinger equation in the modulation space Ms2, q ( R), 1 = q = 2 and s = 0. In addition, for either s = 0 and 1 = q = 3 2 or 3 2 < q = 2 and s > 2 3 - 1 q we show that the Cauchy problem is unconditionally wellposed in Ms2, q ( R). It is done with the use of the differentiation by parts technique which had been previously used in the periodic setting.

关键词： Normal form method modulation spaces Nonlinear Schrodinger Local wellposedness Unconditional uniqueness

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