Details on the distribution co-orbit space $\mathcal H_w^\infty$.

Nikolas Hauschka, Peter Balazs, Lukas Köhldorfer

Session: General

Keywords: Localized frame, co-orbit space, distribution space

TL;DR: We go through the construction of the weighted co-orbit space $H_w^\infty$ in detail and show that it is a Banach space.

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.

Submission Number: 106