Metaplectic Gabor Frames and Symplectic Analysis of Time-Frequency Spaces
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866911759103688704 |
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| author | Cordero, Elena Giacchi, Gianluca |
| author_facet | Cordero, Elena Giacchi, Gianluca |
| contents | We introduce new frames, called \textit{metaplectic Gabor frames}, as natural generalizations of Gabor frames in the framework of metaplectic Wigner distributions. Namely, we develop the theory of metaplectic atoms in a full-general setting and prove an inversion formula for metaplectic Wigner distributions on $\mathbb{R}^d$. Its discretization provides metaplectic Gabor frames.
Next, we deepen the understanding of the so-called shift-invertible metaplectic Wigner distributions, showing that they can be represented, up to chirps, as rescaled short-time Fourier transforms. As an application, we derive a new characterization of modulation and Wiener amalgam spaces. Thus, these metaplectic distributions (and related frames) provide meaningful definitions of local frequencies and can be used to measure effectively the local frequency content of signals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_04593 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Metaplectic Gabor Frames and Symplectic Analysis of Time-Frequency Spaces Cordero, Elena Giacchi, Gianluca Analysis of PDEs Functional Analysis 42C15, 42B35, 42A38 We introduce new frames, called \textit{metaplectic Gabor frames}, as natural generalizations of Gabor frames in the framework of metaplectic Wigner distributions. Namely, we develop the theory of metaplectic atoms in a full-general setting and prove an inversion formula for metaplectic Wigner distributions on $\mathbb{R}^d$. Its discretization provides metaplectic Gabor frames. Next, we deepen the understanding of the so-called shift-invertible metaplectic Wigner distributions, showing that they can be represented, up to chirps, as rescaled short-time Fourier transforms. As an application, we derive a new characterization of modulation and Wiener amalgam spaces. Thus, these metaplectic distributions (and related frames) provide meaningful definitions of local frequencies and can be used to measure effectively the local frequency content of signals. |
| title | Metaplectic Gabor Frames and Symplectic Analysis of Time-Frequency Spaces |
| topic | Analysis of PDEs Functional Analysis 42C15, 42B35, 42A38 |
| url | https://arxiv.org/abs/2304.04593 |