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

TL;DR

This paper rigorously introduces and analyzes the properties of the distribution space al{H}_w^{\u2205} associated with localized frames in Hilbert spaces, clarifying its topological structure and Banach space nature.

Contribution

It provides a rigorous definition and analysis of the weighted distribution space al{H}_w^{\u2205} in a general setting, filling gaps in existing literature.

Findings

al{H}_w^{\u2205} is a Banach space.

Comparison of weak* and norm topologies on al{H}_w^{al{H}}.

Established properties of al{H}_w^{al{H}} and its variants.

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.