# A characterization of the higher dimensional groups associated with continuous wavelets

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DOI: 10.1007/BF02930862

- Cite this article as:
- Laugesen, R.S., Weaver, N., Weiss, G.L. et al. J Geom Anal (2002) 12: 89. doi:10.1007/BF02930862

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## Abstract

A subgroup D of GL (n, ℝ) 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 ∈ 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. ∈ ℝ^{n}; 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 ∈ 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.