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Excursus on modulation spaces via metaplectic operators and related time-frequency representations

Elena Cordero, Gianluca Giacchi

Abstract

Modulation spaces were originally introduced by Feichtinger in 1983. Since the 2000s there have been thousands of contributions using them as correct framework; they range from PDEs, pseudodifferential operators, quantum mechanics, signal analysis. This justifies a deep study of such spaces and the related Wiener ones. Recently, metaplectic Wigner distributions, which contain as special examples the $τ$-Wigner distributions, the ambiguity function and the Short-time Fourier transform, have proved to characterize modulation spaces, under suitable assumptions. We investigate the metaplectic action which is hidden in their construction and guarantees equivalent (quasi-)norms for such spaces. We add a new result on this topic and conclude with an exhaustive vision of these characterizations. Similar results hold for the Wiener amalgam ones.

Excursus on modulation spaces via metaplectic operators and related time-frequency representations

Abstract

Modulation spaces were originally introduced by Feichtinger in 1983. Since the 2000s there have been thousands of contributions using them as correct framework; they range from PDEs, pseudodifferential operators, quantum mechanics, signal analysis. This justifies a deep study of such spaces and the related Wiener ones. Recently, metaplectic Wigner distributions, which contain as special examples the -Wigner distributions, the ambiguity function and the Short-time Fourier transform, have proved to characterize modulation spaces, under suitable assumptions. We investigate the metaplectic action which is hidden in their construction and guarantees equivalent (quasi-)norms for such spaces. We add a new result on this topic and conclude with an exhaustive vision of these characterizations. Similar results hold for the Wiener amalgam ones.
Paper Structure (8 sections, 16 theorems, 75 equations)

This paper contains 8 sections, 16 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weighted mixed norm spaces
  4. Time-frequency analysis tools
  5. Modulation spaces KB2020F1Feichtinger_1981_BanachbookGalperin2004Kobayashi2006PILIPOVIC2004194
  6. The symplectic group $Sp(d,\mathbb{R})$ and the metaplectic operators
  7. Metaplectic Wigner distributions
  8. The submanifold of shift-invertible symplectic matrices

Key Result

Theorem 1.1

Consider $0<p,q\leq\infty$, $\hat{A}\in Mp(d,\mathbb{R})$, $\pi^{Mp}(\hat{A})=A\in Sp(d,\mathbb{R})$. The following statements are equivalent: (i) $\hat{A}: M^{p,q}\to M^{p,q}$ is well defined; (ii) $\hat{A}: M^{p,q}\to M^{p,q}$ is well defined and bounded (in fact, it is an isomorphism); (iii) One

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Proposition 2.1
  • Proposition 2.2
  • ...and 12 more