The Continuous Wavelet Transform and Symmetric Spaces

Authors: Fabec R.¹; Ólafsson G.²

Source: Acta Applicandae Mathematicae, Volume 77, Number 1, May 2003 , pp. 41-69(29)

Publisher: Springer

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Abstract:

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) sime V₁ oplus sdot sdot sdot oplus V_k under the action of the semidirect product H×_sRⁿ. It is well known, that the space V_j contains an admissible vector if and only if the stabilizer in H^t of every point in V_j 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.

Keywords: wavelet; unitary representation; square integrable representation; reproducing Hilbert space; symmetric space

Language: English

Document Type: Research article

Affiliations: 1: Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A. e-mail: fabec@math.lsu.edu 2: Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A. e-mail: olafsson@math.lsu.edu

Publication date: 2003-05-01