The $\alpha$-Modulation Transform: Admissibility, Coorbit Theory and Frames of Compactly Supported Functions

TL;DR

This paper introduces the $\alpha$-modulation transform, establishing new admissibility conditions, applying coorbit theory, and constructing compactly supported frames for $\alpha$-modulation spaces, advancing time-frequency analysis methods.

Contribution

It provides new admissibility criteria, extends coorbit theory to this transform, and constructs compactly supported admissible windows, surpassing previous results.

Findings

Established new admissibility conditions for windows.

Applied coorbit theory to $\alpha$-modulation transform.

Constructed compactly supported admissible windows.

Abstract

The $\alpha$-modulation transform is a time-frequency transform generated by square-integrable representations of the affine Weyl-Heisenberg group modulo suitable subgroups. In this paper we prove new conditions that guarantee the admissibility of a given window function. We also show that the generalized coorbit theory can be applied to this setting, assuming specific regularity of the windows. This then yields canonical constructions of Banach frames and atomic decompositions in $\alpha$-modulation spaces. In particular, we prove the existence of compactly supported (in time domain) vectors that are admissible and satisfy all conditions within the coorbit machinery, which considerably go beyond known results.