Admissible vectors for the regular representation

Abstract

Abstract. It is well known that for irreducible, square-integrable representations of a locally compact group, there exist so-called admissible vectors which allow the construction of generalized continuous wavelet transforms. In this paper we discuss when the irreducibility requirement can be dropped, using a connection between generalized wavelet transforms and Plancherel theory. For unimodular groups with type I regular representation, the existence of admissible vectors is equivalent to a finite measure condition. The main result of this paper states that this restriction disappears in the nonunimodular case: Given a nondiscrete, second countable group G with type I regular representation λ G , we show that λ G itself (and hence every subrepresentation thereof) has an admissible vector in the sense of wavelet theory iff G is nonunimodular.

The density theorem for discrete series representations restricted to lattices

“…The related question as to whether a system (1.1) is a (discrete) frame is at the core of modern frame theory [19] and has, in particular, a long history in Gabor theory [36]. The existence of a frame vector is also studied in representation theory, in whose jargon such a vector is called admissible [25,26]. While the mere existence of a frame or Riesz vector for a given lattice is quite different from the validity of these properties for one specific vector, there is an interesting interplay between the two problems.…”

Admissibility for a Class of Quasiregular Representations

“…By Lemma 2.3, this means that the regular representation of the unimodular group G decomposes into a finite sum of irreducible (square integrable) representations. It is well known (see for example [11,Proposition 0.4]) that this can only happen when G is discrete.…”

“…By virtue of the results [13,Theorem 1.8] and [11,Theorem 0.2], we expect the existence of admissible vectors to be tied to the non-unimodularity of G, and this is shown to be precisely the case. Note that in this context, both H and N are unimodular, so G is non-unimodular if and only if the H-action on N is non-unimodular.…”

“…In the event that W ψ actually defines an isometry of H τ into L 2 (G), then we say that W ψ is a continuous wavelet transform, and that ψ is admissible for τ . When G has Type I reduced dual, the two extreme cases -where τ is irreducible or where τ is the regular representation -are well understood [8,11]. Most closely related to discrete wavelets is the case where G is a semidirect product G = N ⋊ H with N normal and where τ is the quasiregular representation of G in L 2 (N).…”

CONTINUOUS ACTION OF LIE GROUPS ON ℝn AND FRAMES

“…Since then several people have worked on wavelets related to actions of topological groups acting on R n . Without trying to be complete we would like to name the work of Ali, Antoine, and Gazeau, [1,2], Bernier and Taylor [4], Für and Führ and Mayer [11,12,13,14], and finally Laugesen, Weaver, Weiss, and Wilson [19]. In most of these cases the group generalizing the (ax + b)-group is a semidirect product R n × s H, where H is a closed subgroup of GL(n, R).…”