$\mathcal{A}$-Localization Operators

TL;DR

This paper introduces $\\mathcal{A}$-localization operators, a broad generalization of classical time-frequency localization operators, linking metaplectic Wigner distributions with pseudodifferential calculus and analyzing their properties.

Contribution

It extends the relation between localization operators and Weyl quantization to covariant metaplectic Wigner distributions, providing a unified framework for quantization and signal analysis.

Findings

Established boundedness on modulation spaces

Provided conditions for Schatten-von Neumann class membership

Unified approach to quantization and 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 \[ W_\mathcal{A}(\pi(z)f,\pi(z)g)=T_zW_\mathcal{A}(f,g), \qquad z\in\mathbb{R}^{2d}, \] 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 $\tau$-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.