The continuous wavelet transform and symmetric spaces

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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) ≃ V1 ⊕...⊕ Vk 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.

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Acta Applicandae Mathematicae

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