The Journal of Geometric Analysis Volume 12, Number 1, 2002 A Characterization of the Higher Dimensional Groups Associated with Continuous Wavelets By R.S. Laugesen, N. Weaver, G.L. Weiss, and E.N. Wilson ABSTRACT. A subgroup D of G L(n, R) is said to be admissible if the semidirect product of D and ~ n, considered as a subgroup of the affine group on R n, admits wavelets ~t ~ L 2 (Rn ) satisfying a generalization of the Calder6n reproducing formula. This article provides a nearly complete characterization of the admissible subgroups D. More precisel); if D is admissible, then the stability subgroup Dx for the transpose action of D on R n must be compact for a. e. x E ~n ; moreover, if A is the modular function of D, there must exist an a ~ D such that Idetal # A(a). Conversely, if the last condition holds and for a. e. x ~ Nn there exists an E > Ofor which the E-stabilizer D~x is compact, then D is admissible. Numerous examples are given of both admissible and non-admissible groups. 1. Introduction In order to introduce the wavelets we shall consider and the groups associated with them, it is useful to begin with some one dimensional results A.P. Calder6n introduced in 1964, [5]. Let G be the affine group associated with N. That is, G consists of all (a, b) E R • N, a ~ 0, with the group operation (c, d) . (a, b) = (ca, b + d) 9 This operation is consistent with the action of s = (a, b) ~ G on x ~ N given by s (x) = a (x + b). For 7t E L2(~) let 1 (X-b)=l ( ) (r~) (x) = ~ ~ a ~ ~ s-l(x) -- ~a,b(X). Then s w-> Ts is a unitary representation of G acting on L2(R). The mapping W0, taking f 6 Lz(N) into the bounded function on G defined by (Wo f ) (s) = [ f (x) (rs ap )(x) dx = (f , 1/fa,b) JR Math Subject Classifications. 42MCS2000. Key Words and Phrases. continuous wavelets, discrete wavelets, admissable dialatin groups. Acknowledgements and Notes. The first named author was supported by NSF Grant DMS-9970228. The third named author was supported by the South Western Bell Co. 9 2002TheJournal of Geometric Analysis ISSN 1050-6926