Wavelet Transforms for Semidirect Product Groups with Not Necessarily Commutative Normal Subgroups

Ishi, Hideyuki

doi:10.1007/s00041-005-5002-0

Cited by 10 publications

(4 citation statements)

References 21 publications

(25 reference statements)

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“…The construction of wavelets on homogeneous groups has recently been studied by various authors [88,174,181,233,302]. The existence of such functions was observed in [174].…”

Section: As Well As By Wavelet Inversionmentioning

confidence: 99%

“…The existence of such functions was observed in [174]. However, the purely representation-theoretic techniques of [174] did not allow to establish the existence of "nice" wavelet functions, with suitable smoothness and decay properties, and similar comments apply to the arguments in [88,233]. The first nice wavelet to be constructed for this context was the Mexican Hat wavelet on the Heisenberg group, introduced by Mayeli [302,303], though by a somewhat more intricate argument.…”

Section: As Well As By Wavelet Inversionmentioning

confidence: 99%

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Mathematical Methods for Signal and Image Analysis and Representation

Florack¹,

Duits²,

Jongbloed³

et al. 2011

Computational Imaging and Vision

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This comprehensive book series embraces state-of-the-art expository works and advanced research monographs on any aspect of this interdisciplinary field.Topics covered by the series fall in the following four main categories:• Imaging Systems and Image Processing • Computer Vision and Image Understanding • Visualization •Applications of Imaging TechnologiesOnly monographs or multi-authored books that have a distinct subject area, that is where each chapter has been invited in order to fulfill this purpose, will be considered for the series. Volume 41 For further volumes: www.springer.com/series/5754 Luc Florack r Remco Duits r Geurt Jongbloed r Marie-Colette van Lieshout r Laurie Davies Editors

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“…The construction of wavelets on homogeneous groups has recently been studied by various authors [88,174,181,233,302]. The existence of such functions was observed in [174].…”

Section: As Well As By Wavelet Inversionmentioning

confidence: 99%

Section: As Well As By Wavelet Inversionmentioning

confidence: 99%

Mathematical Methods for Signal and Image Analysis and Representation

Florack¹,

Duits²,

Jongbloed³

et al. 2011

Computational Imaging and Vision

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“…Several authors developed the theory of continuous wavelet transform on the Heisenberg group H n (see [16], [22]). Recently, some further extensions of wavelet analysis were published in [10], [18]. The Radon transform represents an interesting object from the point of view of both harmonic analysis and integral geometry.…”

Section: Introductionmentioning

confidence: 99%

Wavelet transform and radon transform on the quaternion Heisenberg group

Liu

2014

Acta. Math. Sin.-English Ser.

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Let Q be the quaternion Heisenberg group, and let P be the affine automorphism group of Q. We develop the theory of continuous wavelet transform on the quaternion Heisenberg group via the unitary representations of P on L 2 (Q). A class of radial wavelets is constructed. The inverse wavelet transform is simplified by using radial wavelets. Then we investigate the Radon transform on Q. A Semyanistri-Lizorkin space is introduced, on which the Radon transform is a bijection. We deal with the Radon transform on Q both by the Euclidean Fourier transform and the group Fourier transform. These two treatments are essentially equivalent. We also give an inversion formula by using wavelets, which does not require the smoothness of functions if the wavelet is smooth.

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“…A generalization of this latter situation is studied in [5], where N is an arbitrary simply connected nilpotent group, and H is a vector group acting on N by R-split semi-simple automorphisms. Finally, in a somewhat different direction, it is shown in [11] that τ has admissible vectors when the dualN is (a.e.) a finite union of H-orbits, and this occurs in a number of important situations, for example [7].…”

Section: Introductionmentioning

confidence: 99%

Decomposition and admissibility for the quasiregular representation for generalized oscillator groups

Currey¹,

McNamara²

2008

Contemporary Mathematics

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Abstract. We analyze the co-normally induced quasiregular representation for two families of Lie groups: the 2d-oscillator groups N SO(2d) where N is the free two-step nilpotent group on 2d generators, and the dilated 2d-oscillator groups N (SO(2d) × R * + ). We construct irreducible decompositions in both cases with explicit spectrum and intertwining operators, and in both cases we prove a Caldéron-type admissibility condition for multiplicity-free, quasiequivalent subrepresentations. We prove that in the case of the 2d-oscillator groups, the quasiregular representation has no admissible vectors, and for the dilated 2d-oscillator groups, we give an explicit construction for admissible vectors.

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Wavelet Transforms for Semidirect Product Groups with Not Necessarily Commutative Normal Subgroups

Cited by 10 publications

References 21 publications

Mathematical Methods for Signal and Image Analysis and Representation

Mathematical Methods for Signal and Image Analysis and Representation

Wavelet transform and radon transform on the quaternion Heisenberg group

Decomposition and admissibility for the quasiregular representation for generalized oscillator groups

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