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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2501.12845 |
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| _version_ | 1866909469124853760 |
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| author | Biswas, Md Hasan Ali Thangavelu, Sundaram |
| author_facet | Biswas, Md Hasan Ali Thangavelu, Sundaram |
| contents | In this article we show how certain irreducible unitary representation $ Π_λ$ of the twisted Heisenberg group $ \He_λ^n(\C)$ leads to the twisted modulation spaces $ M_λ^{p,q}(\R^{2n}).$ These $ Π_λ$ also turn out to be irreducible unitary representations of another nilpotent Lie group $ G_n $ which contains two copies of the Heisenberg group $ \He^n.$ By lifting $ Π_λ$ we obtain another unitary representation $ Π$ of $ G_n $ acting on $ L^2(\He^n).$ We define our modulation spaces $ M^{p,q}(\He^n) $ in terms of the matrix coefficients associated to $ Π.$ These spaces are shown to be invariant under Heisenberg translations and Heisenberg modulations which are different from euclidean modulations. We also establish some of the basic properties of $ M_λ^{p,q}(\R^{2n})$ and $ M^{p,q}(\He^n) $ such as completeness and invariance under suitable Fourier transforms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_12845 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Modulation spaces on the Heisenberg group Biswas, Md Hasan Ali Thangavelu, Sundaram Functional Analysis Representation Theory In this article we show how certain irreducible unitary representation $ Π_λ$ of the twisted Heisenberg group $ \He_λ^n(\C)$ leads to the twisted modulation spaces $ M_λ^{p,q}(\R^{2n}).$ These $ Π_λ$ also turn out to be irreducible unitary representations of another nilpotent Lie group $ G_n $ which contains two copies of the Heisenberg group $ \He^n.$ By lifting $ Π_λ$ we obtain another unitary representation $ Π$ of $ G_n $ acting on $ L^2(\He^n).$ We define our modulation spaces $ M^{p,q}(\He^n) $ in terms of the matrix coefficients associated to $ Π.$ These spaces are shown to be invariant under Heisenberg translations and Heisenberg modulations which are different from euclidean modulations. We also establish some of the basic properties of $ M_λ^{p,q}(\R^{2n})$ and $ M^{p,q}(\He^n) $ such as completeness and invariance under suitable Fourier transforms. |
| title | Modulation spaces on the Heisenberg group |
| topic | Functional Analysis Representation Theory |
| url | https://arxiv.org/abs/2501.12845 |