Details on the distribution co-orbit space $\mathcal{H}^{\infty}_w$
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866910822118195200 |
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| author | Hauschka, Nikolas Balazs, Peter Köhldorfer, Lukas |
| author_facet | Hauschka, Nikolas Balazs, Peter Köhldorfer, Lukas |
| contents | Associated with every separable Hilbert space $\mathcal{H}$ and a given localized frame, there exists a natural test function Banach space $\mathcal{H}^1$ and a Banach distribution space $\mathcal{H}^{\infty}$ so that $\mathcal{H}^1 \subset \mathcal{H} \subset \mathcal{H}^{\infty}$. In this article we close some gaps in the literature and rigorously introduce the space $\mathcal{H}^{\infty}$ and its weighted variants $\mathcal{H}_w^{\infty}$ in a slightly more general setting and discuss some of their properties. In particular, we compare the underlying weak$^*$- with the norm topology associated with $\mathcal{H}_w^{\infty}$ and show that $(\mathcal{H}_w^{\infty}, \Vert \cdot \Vert_{\mathcal{H}_w^{\infty}})$ is a Banach space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_07378 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Details on the distribution co-orbit space $\mathcal{H}^{\infty}_w$ Hauschka, Nikolas Balazs, Peter Köhldorfer, Lukas Functional Analysis Associated with every separable Hilbert space $\mathcal{H}$ and a given localized frame, there exists a natural test function Banach space $\mathcal{H}^1$ and a Banach distribution space $\mathcal{H}^{\infty}$ so that $\mathcal{H}^1 \subset \mathcal{H} \subset \mathcal{H}^{\infty}$. In this article we close some gaps in the literature and rigorously introduce the space $\mathcal{H}^{\infty}$ and its weighted variants $\mathcal{H}_w^{\infty}$ in a slightly more general setting and discuss some of their properties. In particular, we compare the underlying weak$^*$- with the norm topology associated with $\mathcal{H}_w^{\infty}$ and show that $(\mathcal{H}_w^{\infty}, \Vert \cdot \Vert_{\mathcal{H}_w^{\infty}})$ is a Banach space. |
| title | Details on the distribution co-orbit space $\mathcal{H}^{\infty}_w$ |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2502.07378 |