The Continuous Wavelet Transform and Symmetric Spaces
Source: Acta Applicandae Mathematicae, Volume 77, Number 1, May 2003 , pp. 41-69(29)
The continuous wavelet transform has become a widely used tool in applied science during the last decade. In this article we discuss some generalizations coming from actions of closed subgroups H of GL(n,R) acting on Rn. If Rn has finitely many open orbits under the transposed action of H such that the union has full measure, then L2(Rn) decomposes into finitely many irreducible representations, L2(Rn)V1Vk under the action of the semidirect product H×sRn. It is well known, that the space Vj contains an admissible vector if and only if the stabilizer in Ht of every point in Vj is compact. In this article we discuss the case where the stabilizer of a generic point in Rn is not compact, but a symmetric subgroup, a case that has not previously been discussed in the literature. In particular we show, that the wavelet transform can always be inverted in this case.
Document Type: Research article
Affiliations: 1: Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A. e-mail: firstname.lastname@example.org 2: Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A. e-mail: email@example.com
Publication date: 2003-05-01