A characterization of the higher dimensional groups associated with continuous wavelets

A subgroup D of GL (n, ℝ) is said to be admissible if the semidirect product of D and ℝn, considered as a subgroup of the affine group on ℝn, admits wavelets ψ ∈ L2(ℝn) 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 Dx for the transpose action of D on ℝn must be compact for a. e. x. ∈ ℝn; moreover, if Δ is the modular function of D, there must exist an a ∈ D such that |det a| ≠ Δ(a). Conversely, if the last condition holds and for a. e. x ∈ ℝn there exists an ε > 0 for which the ε-stabilizer Dxε is compact, then D is admissible. Numerous examples are given of both admissible and non-admissible groups.

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