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$\mathcal{A}$-Localization Operators

Elena Cordero, Edoardo Pucci

TL;DR

This work generalizes time-frequency localization operators by embedding them in the metaplectic framework through covariant metaplectic Wigner distributions W_{\mathcal{A}}. It proves a representation formula A_{a}^{φ1,φ2} = Op_{\mathcal{A}}(a * W_{\mathcal{A}}(φ2,φ1)) that characterizes covariance, and defines A_{a,\mathcal{A}}^{φ1,φ2} as a universal localization operator. The paper then derives Schwartz kernels, explicit kernels for totally-Wigner-decomposable A, and sharp boundedness and Schatten-class results on modulation spaces, including adjoint relations and L^p-continuity under structural block conditions. By unifying localization, quantization, and signal analysis within the metaplectic setting, it provides a broad, structurally coherent framework for phase-space localization and operator quantization with practical implications for time-frequency analysis.

Abstract

Time-frequency localization operators, originally introduced by Daubechies (1988), provide a framework for localizing signals in the phase space and have become a central tool in time-frequency analysis. In this paper we introduce and study a broad generalization of these operators, called $\mathcal{A}$-localization operators, associated with a metaplectic Wigner distribution $W_\mathcal{A}$ and the corresponding $\mathcal{A}$-pseudodifferential calculus. We first show that the classical relation between localization operators and Weyl quantization extends to any \emph{covariant metaplectic Wigner distribution}. Specifically, if $W_\mathcal{A}$ satisfies the covariance property \[ W_\mathcal{A}(π(z)f,π(z)g)=T_zW_\mathcal{A}(f,g), \qquad z\in\mathbb{R}^{2d}, \] then \[ A_{a}^{\varphi_1,\varphi_2} = \operatorname{Op}_\mathcal{A}\big(a * W_\mathcal{A}(\varphi_2,\varphi_1)\big), \] and conversely, this identity characterizes covariance. This result extends the recent representation formula of Bastianoni and Teofanov for $τ$-operators to the full metaplectic framework. We then define the $\mathcal{A}$-localization operator $A_{a,\mathcal{A}}^{\varphi_1,\varphi_2}$ and investigate its analytical properties. We establish boundedness results on modulation spaces and provide sufficient conditions for Schatten-von Neumann class membership. These findings connect the structure of metaplectic representations with time-frequency localization theory, offering a unified approach to quantization and signal analysis.

$\mathcal{A}$-Localization Operators

TL;DR

This work generalizes time-frequency localization operators by embedding them in the metaplectic framework through covariant metaplectic Wigner distributions W_{\mathcal{A}}. It proves a representation formula A_{a}^{φ1,φ2} = Op_{\mathcal{A}}(a * W_{\mathcal{A}}(φ2,φ1)) that characterizes covariance, and defines A_{a,\mathcal{A}}^{φ1,φ2} as a universal localization operator. The paper then derives Schwartz kernels, explicit kernels for totally-Wigner-decomposable A, and sharp boundedness and Schatten-class results on modulation spaces, including adjoint relations and L^p-continuity under structural block conditions. By unifying localization, quantization, and signal analysis within the metaplectic setting, it provides a broad, structurally coherent framework for phase-space localization and operator quantization with practical implications for time-frequency analysis.

Abstract

Time-frequency localization operators, originally introduced by Daubechies (1988), provide a framework for localizing signals in the phase space and have become a central tool in time-frequency analysis. In this paper we introduce and study a broad generalization of these operators, called -localization operators, associated with a metaplectic Wigner distribution and the corresponding -pseudodifferential calculus. We first show that the classical relation between localization operators and Weyl quantization extends to any \emph{covariant metaplectic Wigner distribution}. Specifically, if satisfies the covariance property then and conversely, this identity characterizes covariance. This result extends the recent representation formula of Bastianoni and Teofanov for -operators to the full metaplectic framework. We then define the -localization operator and investigate its analytical properties. We establish boundedness results on modulation spaces and provide sufficient conditions for Schatten-von Neumann class membership. These findings connect the structure of metaplectic representations with time-frequency localization theory, offering a unified approach to quantization and signal analysis.

Paper Structure

This paper contains 13 sections, 17 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Schatten-von Neumann Classes
  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. Metaplectic pseudodifferential operators
  9. $\mathcal{A}$-localization operators
  10. Counterexamples for non-covariant matrices $\mathcal{A}$
  11. Schwartz kernel of $A_{a,\mathcal{A}}^{\varphi_1,\varphi_2}$
  12. Totally-Wigner decomposable $\mathcal{A}\in Sp(2d,\mathbb{R})$
  13. Continuity properties

Key Result

Theorem 1.1

Let $\varphi_1,\varphi_2\in\mathcal{S}(\mathbb{R}^d)\setminus\{0\}$, $a\in\mathcal{S}'(\mathbb{R}^{2d})$, and $\mathcal{A}\in \mathrm{Sp}(2d,\mathbb{R})$. If $W_\mathcal{A}$ is covariant, then Conversely, if this identity holds for all $a\in\mathcal{S}'({\mathbb{R}^{2d}}),\varphi_1,\varphi_2\in\mathcal{S}(\mathbb{R}^d)$, then $W_\mathcal{A}$ is covariant.

Theorems & Definitions (48)

  • Theorem 1.1
  • Remark 2.1
  • Example 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Proposition 2.5: Proposition 4.4 in CR2021
  • Definition 2.6
  • Remark 2.7
  • Example 2.8
  • Lemma 2.9: Lemma 3.2. in CGR2022
  • ...and 38 more