Characterizing Abelian Admissible Groups

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 for checking admissibility of a matrix group generated by finitely many commuting matrices with positive spectra. We also present examples showing that the simple answers that are available for the real spectrum case fail in the general case. An interesting byproduct of our considerations is a method that allows for an abelian Lie subalgebra 𝔥⊂ gl(n,ℝ) to check whether H = exp(𝔥) is closed.