On a discrete transform of homogeneous decomposition spaces

TL;DR

This paper introduces almost diagonal matrices in discrete homogeneous function spaces, demonstrating their closure under multiplication and their role in frame transformations, leading to compact frame expansions for these spaces.

Contribution

It develops the concept of almost diagonal matrices in discrete homogeneous spaces and links them to continuous frame transformations, enabling new frame expansion constructions.

Findings

Almost diagonal matrices form a closed algebra under multiplication.

Change of frame matrices with decay are almost diagonal.

Constructs compactly supported frame expansions for homogeneous spaces.

Abstract

We introduce almost diagonal matrices in the setting of (anisotropic) discrete homogeneous Triebel-Lizorkin type spaces and homogeneous modulation spaces, and it is shown that the class of almost diagonal matrices is closed under matrix multiplication. We then connect the results to the continuous setting and show that the "change of frame" matrix for a pair of time-frequency frames, with suitable decay properties, is almost diagonal. As an application of this result, we consider a construction of compactly supported frame expansions for homogeneous decomposition spaces of Triebel-Lizorkin type and for the associated modulation spaces.