Non-isotropic translation and modulation invariant Hilbert spaces

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 \xi}f(\cdo -x)}{\mathcal H}\le C_0\nm f{\mathcal H}, \qquad f\in \mathcal H ,\ x,...

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Hauptverfasser:	Ratnakumar, P. K, Toft, Joachim, Vindas, Jasson
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Schlagworte:	Mathematics - Functional Analysis
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creator	Ratnakumar, P. K Toft, Joachim Vindas, Jasson
description	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 $$ \nm {e^{i\scal \cdo \xi}f(\cdo -x)}{\mathcal H}\le C_0\nm f{\mathcal H}, \qquad f\in \mathcal H ,\ x,\xi \in \mathbf R^d, $$ then we prove that $\maclH = L^2(\mathbf R^d)$, with equivalent norms.
doi_str_mv	10.48550/arxiv.2407.08435
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title	Non-isotropic translation and modulation invariant Hilbert spaces
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