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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.00671 |
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| _version_ | 1866918181795266560 |
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| author | Cordero, Elena Pucci, Edoardo |
| author_facet | Cordero, Elena Pucci, Edoardo |
| contents | 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
\[
A_{a}^{φ_1,φ_2}
= \operatorname{Op}_\mathcal{A}\big(a * W_\mathcal{A}(φ_2,φ_1)\big),
\]
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 $A_{a,\mathcal{A}}^{φ_1,φ_2}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_00671 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $\mathcal{A}$-Localization Operators Cordero, Elena Pucci, Edoardo Functional Analysis 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 \[ A_{a}^{φ_1,φ_2} = \operatorname{Op}_\mathcal{A}\big(a * W_\mathcal{A}(φ_2,φ_1)\big), \] 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 $A_{a,\mathcal{A}}^{φ_1,φ_2}$ 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. |
| title | $\mathcal{A}$-Localization Operators |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2511.00671 |