Admissibility for $\alpha$-Modulation Spaces

TL;DR

This paper advances the theory of $\alpha$-modulation spaces by establishing new admissibility conditions, enabling the construction of compactly supported atoms for frame decompositions, which benefits numerical applications.

Contribution

It introduces improved admissibility conditions for $\alpha$-modulation spaces, ensuring the existence of compactly supported admissible vectors for better numerical implementations.

Findings

New admissibility conditions for $\alpha$-modulation spaces

Existence of compactly supported admissible vectors

Enhanced framework for frame decompositions

Abstract

This paper is concerned with frame decompositions of $\alpha$-modulation spaces. These spaces can be obtained as coorbit spaces for square-integrable representations of the affine Weyl-Heisenberg group modulo suitable subgroups. The theory yields canonical constructions for Banach frames or atomic decompositions in $\alpha$-modulation spaces. A necessary ingredient in this abstract machinery is the existence of generating functions that are admissible for the representation. For numerical purposes, admissible atoms with compact support are necessary. We show new admissibility conditions that considerably improve upon known results. In particular, we prove the existence of admissible vectors that have compact support in time domain.