A characterization of the higher dimensional groups associated with continuous wavelets

A subgroup D of GL (n, <) is said to be admissible if the semidirect product of D and ℝ ⁿ, considered as a subgroup of the affine group on ℝ ⁿ, admits wavelets ψ ε L ²(ℝ ⁿ ) satisfying a generalization of the Calderón reproducing, formula. This article provides a nearly complete characterization of the admissible subgroups D. More precisely, if D is admissible, then the stability subgroup D _x for the transpose action of D on ℝ ⁿ must be compact for a. e. x ε ℝ ⁿ; moreover, if Δ is the modular function of D, there must exist an a ε D such that |det a| ≠ Δ(a). Conversely, if the last condition holds and for a. e. x ε ℝ ⁿ there exists an ε > 0 for which the ε-stabilizer D _x ^ε is compact, then D is admissible. Numerous examples are given of both admissible and non-admissible groups.