Twisted shift preserving operators on $L^{2}(\mathbb{R}^{2n})$
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909197195542528 |
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| author | Velsamy, Rabeetha Ramakrishnan, Radha |
| author_facet | Velsamy, Rabeetha Ramakrishnan, Radha |
| contents | We introduce the J map using the Zak transform associated with the Weyl transform on $L^{2}(\mathbb{R}^{2n})$. We obtain a decomposition for a twisted shift-invariant subspace of $L^{2}(\mathbb{R}^{2n})$ as a direct sum of mutually orthogonal principal twisted shift-invariant spaces such that the respective system of twisted translates forms a Parseval frame sequence. We establish that the twisted shift preserving operators and the corresponding range operators simultaneously share some properties in common, namely, self-adjoint, unitary, range of the spectrum and bounded below properties. We prove that the frame operator and its inverse associated with a system of twisted translates of {φ_s}_{s\in Z} are shift preserving. We also show that the corresponding range operators turn out to be the dual Gramian and its inverse associated with the collection {Jφ_s(. , .)}_{s\in Z}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_13238 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Twisted shift preserving operators on $L^{2}(\mathbb{R}^{2n})$ Velsamy, Rabeetha Ramakrishnan, Radha Functional Analysis Primary 42C15, Secondary 42B10, 47B02 We introduce the J map using the Zak transform associated with the Weyl transform on $L^{2}(\mathbb{R}^{2n})$. We obtain a decomposition for a twisted shift-invariant subspace of $L^{2}(\mathbb{R}^{2n})$ as a direct sum of mutually orthogonal principal twisted shift-invariant spaces such that the respective system of twisted translates forms a Parseval frame sequence. We establish that the twisted shift preserving operators and the corresponding range operators simultaneously share some properties in common, namely, self-adjoint, unitary, range of the spectrum and bounded below properties. We prove that the frame operator and its inverse associated with a system of twisted translates of {φ_s}_{s\in Z} are shift preserving. We also show that the corresponding range operators turn out to be the dual Gramian and its inverse associated with the collection {Jφ_s(. , .)}_{s\in Z}. |
| title | Twisted shift preserving operators on $L^{2}(\mathbb{R}^{2n})$ |
| topic | Functional Analysis Primary 42C15, Secondary 42B10, 47B02 |
| url | https://arxiv.org/abs/2308.13238 |