Non-isometric translation and modulation invariant Hilbert spaces
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866918051006382080 |
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| author | Ratnakumar, P. K. Toft, Joachim Vindas, Jasson |
| author_facet | Ratnakumar, P. K. Toft, Joachim Vindas, Jasson |
| contents | Let $\mathcal H$ be a Hilbert space of distributions on $\mathbf R^d$ which contains at least one non-zero element in $\mathscr D '(\mathbf R^d)$. If there is a constant $C_0>0$ such that $$ \nm {e^{i\scal \cdo ξ}f(\cdo -x)}{\mathcal H}\le C_0\nm f{\mathcal H}, \qquad f\in \mathcal H ,\ x,ξ\in \mathbf R^d, $$ then we prove that $\maclH = L^2(\mathbf R^d)$, with equivalent norms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_08435 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-isometric translation and modulation invariant Hilbert spaces Ratnakumar, P. K. Toft, Joachim Vindas, Jasson Functional Analysis Let $\mathcal H$ be a Hilbert space of distributions on $\mathbf R^d$ which contains at least one non-zero element in $\mathscr D '(\mathbf R^d)$. If there is a constant $C_0>0$ such that $$ \nm {e^{i\scal \cdo ξ}f(\cdo -x)}{\mathcal H}\le C_0\nm f{\mathcal H}, \qquad f\in \mathcal H ,\ x,ξ\in \mathbf R^d, $$ then we prove that $\maclH = L^2(\mathbf R^d)$, with equivalent norms. |
| title | Non-isometric translation and modulation invariant Hilbert spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2407.08435 |