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arXiv · 2502.07378

Details on the distribution co-orbit space $\mathcal{H}^{\infty}_w$

Nikolas Hauschka·Peter Balazs·Lukas Köhldorfer
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Abstract

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.

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Nikolas Hauschka, Peter Balazs, Lukas Köhldorfer. 2025-02-11. Details on the distribution co-orbit space $\mathcal{H}^{\infty}_w$. https://arxiv.org/abs/2502.07378

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