Continuous Wavelet Transforms from Semidirect Products: Cyclic Representations and Plancherel Measure

Führ, Hartmut; Mayer, Matthias

doi:10.1007/s00041-002-0018-1

Cited by 35 publications

(40 citation statements)

References 26 publications

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“…That we have indeed computed the Plancherel measure is due to [17,Theorem 3.3]. (An alternative argument could be derived from [24, II, Theorem 2.3], or rather, the proof of that result.)…”

Section: Example Of the (1+1)-poincaré Groupmentioning

confidence: 99%

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Plancherel Inversion as Unified Approach to Wavelet Transforms and Wigner Functions

Ali

Führ

Krasowska

2003

Ann. Henri Poincaré

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We demonstrate that the Plancherel transform for Type-I groups provides one with a natural, unified perspective for the generalized continuous wavelet transform, on the one hand, and for a class of Wigner functions, on the other. We first prove that a Plancherel inversion formula, well known for Bruhat functions on the group, holds for a much larger class of functions. This result allows us to view the wavelet transform as essentially the inverse Plancherel transform. The wavelet transform of a signal is an L 2 -function on an appropriately chosen group while the Wigner function is defined on a coadjoint orbit of the group and serves as an alternative characterization of the signal, which is often used in practical applications. The Plancherel transform maps L 2 -functions on a group unitarily to fields of Hilbert-Schmidt operators, indexed by unitary irreducible representations of the group. The wavelet transform can essentially be looked upon as a restricted inverse Plancherel transform, while Wigner functions are modified Fourier transforms of inverse Plancherel transforms, usually restricted to a subset of the unitary dual of the group. Some known results on both Wigner functions and wavelet transforms, appearing in the literature from very different perspectives, are naturally unified within our approach. Explicit computations on a number of groups illustrate the theory.

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“…That we have indeed computed the Plancherel measure is due to [17,Theorem 3.3]. (An alternative argument could be derived from [24, II, Theorem 2.3], or rather, the proof of that result.)…”

Section: Example Of the (1+1)-poincaré Groupmentioning

confidence: 99%

“…In our case, the little fixed groups are trivial, and in this case the last step reduces to a -still somewhat subtle -normalization issue. (This is discussed at length in [17]. )…”

Section: Example Of the (1+1)-poincaré Groupmentioning

confidence: 99%

Plancherel Inversion as Unified Approach to Wavelet Transforms and Wigner Functions

Ali

Führ

Krasowska

2003

Ann. Henri Poincaré

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“…A family of examples are the semidirect products G = R n H, and their quasi-regular representations already mentioned in the introduction, and studied, with increasing generality, in the papers [19,13,15,4,10,16]. In these cases both the decomposition of the quasi-regular representation and the Plancherel theory of G can be computed explicitly, and it is instructive to view the various admissibility conditions derived for those groups in the light of the abstract approach; see [11] for a detailed exposition.…”

Section: Hartmut Führmentioning

confidence: 99%

Admissible vectors for the regular representation

Führ¹

2002

Proc. Amer. Math. Soc.

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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.

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“…(7) holds. However, the following example, adapted from [14] and [15], shows that A 0 can be unbounded (compare with the notions of weak and strong square-integrability in [14]). …”

Section: Corollary 2 With the Notation Of Theorem 1 Letmentioning

confidence: 99%

Square-integrability modulo a subgroup

Cassinelli

Vito

2002

Trans. Amer. Math. Soc.

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Abstract. We prove a weak form of the Frobenius reciprocity theorem for locally compact groups. As a consequence, we propose a definition of squareintegrable representation modulo a subgroup that clarifies the relations between coherent states, wavelet transforms and covariant localisation observables. A self-contained proof of the imprimitivity theorem for covariant positive operator-valued measures is given.

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Continuous Wavelet Transforms from Semidirect Products: Cyclic Representations and Plancherel Measure

Cited by 35 publications

References 26 publications

Plancherel Inversion as Unified Approach to Wavelet Transforms and Wigner Functions

Plancherel Inversion as Unified Approach to Wavelet Transforms and Wigner Functions

Admissible vectors for the regular representation

Square-integrability modulo a subgroup

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