Characterizing abelian admissible groups

TL;DR

This paper establishes necessary and sufficient conditions for the admissibility of abelian matrix groups, facilitating wavelet analysis by providing a clear criterion for such groups.

Contribution

It derives a general admissibility criterion for abelian matrix groups, especially those with real spectra, simplifying the verification process.

Findings

Provides a block diagonalization for commuting matrices

Reduces admissibility decision to connected, simply connected groups

Offers an easily checked criterion for abelian groups with real spectra

Abstract

By definition, admissible matrix groups are those that give rise to a wavelet-type inversion formula. This paper investigates necessary and sufficient admissibility conditions for abelian matrix groups. We start out by deriving a block diagonalization result for commuting real valued matrices. We then reduce the question of deciding admissibility to the subclass of connected and simply connected groups, and derive a general admissibility criterion for exponential solvable matrix groups. For abelian matrix groups with real spectra, this yields an easily checked necessary and sufficient characterization of admissibility. As an application, we sketch a procedure how to check admissibility of a matrix group generated by finitely many commuting matrices with positive spectra.