A characterization of the higher dimensional groups associated with continuous wavelets
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A subgroup D of GL (n, ℝ) is said to be admissible if the semidirect product of D and ℝ n , considered as a subgroup of the affine group on ℝ n , admits wavelets ψ ∈ L2(ℝ n ) satisfying a generalization of the Calderón reproducing, formula. This article provides a nearly complete characterization of the admissible subgroups D. More precisely, if D is admissible, then the stability subgroup Dx for the transpose action of D on ℝ n must be compact for a. e. x. ∈ ℝ n ; moreover, if Δ is the modular function of D, there must exist an a ∈ D such that |det a| ≠ Δ(a). Conversely, if the last condition holds and for a. e. x ∈ ℝ n there exists an ε > 0 for which the ε-stabilizer D x ε is compact, then D is admissible. Numerous examples are given of both admissible and non-admissible groups.
- Ali, S. T., Antoine, J.-P., and Gazeau, J.-P.,Coherent States, Wavelets and Their Generalizations, Springer-Verlag, Inc., New York, (2000).
- Bacry, H., Grossman, A., and Zak, J.Geometry of Generalized Coherent States, Group Theoretical Methods in Physics (Fourth International Colloquium, Nijmegen, 1975), Springer-Verlag, Berlin, Lecture Notes in Physics,50, 249–268, (1978).
- Bernier, D. and Taylor, K.F. Wavelets from square-integrable representations,SIAM J. Math. Anal.,27(2), 594–608, (1996). CrossRef
- Bruna, J. Sampling in complex and harmonic analysis,Proc. of 3 rd Eur. Conf. in Mathematics, July, (2000).
- Calderón, A.P. Intermediate spaces and interpolation, the complex method,Stud. Math.,24, 113–190, (1964).
- Carey, A.L. Square integrable representations of non-unimodular groups,Bull. Austral. Math. Soc.,15, 1–12, (1976). CrossRef
- Chui, C., Czaja, W., Maggioni, M., and Weiss, G.L. Characterization of general tight wavelet frames with matrix dilation and tightness preserving oversampling, to appear inJ. Four. Anal. and Applications.
- Corbett, J. Coherent States on Kinematic Groups: The Study of Spatio-Temporal Wavelets and Their Applications to Motion Estimation, Ph.D. Thesis, Washington University, (1999).
- Duflo, M. and Moore, C.C. On the representation of a nonunimodular locally compact group.J. Funct. Anal.,21, 209–243, (1976). CrossRef
- DeMari, F. and Nowak, K. Reproducing formulas and commutative operator algebras related to affine transformations of the time-frequency plane, preprint.
- Führ, H., Wavelet frames and admissibility in higher dimensions,J. of Math. Phys.,37, 6353–6366, (1996). CrossRef
- Führ, H. Continuous wavelet transforms from semidirect products, to appear inRevista Ciencias Matematicas.
- Führ, H. Admissible vectors for the regular representations, to appear inProc. Am. Math. Soc.
- Führ, H. and Mayer, M. Continuous wavelet transforms from cyclic representations: A general approach using plancherel measure, preprint, (1999).
- Heil, C.E. and Walnut, D.F. Continuous and discrete wavelet transforms,SIAM Review,31, 628–666, (1989). CrossRef
- Isham, C.J. and Klauder, J.R. Coherent states forn-dimensional Euclidean groups and their application,J. of Math. Phys.,32, 607–620, (1991). CrossRef
- Kalisa, C. Etats Cohérents Affines: Canoniques, Galiléns et Relativistes, Ph.D. Thesis, Université Louvaine, (1993).
- Liu, H. and Peng, L. Admissible wavelets associated with the Heisenberg group,Pacific J. of Math.,180, 101–123, (1997). CrossRef
- Weiss, G.L. and Wilson, E.N.The Mathematical Theory of Wavelets, Proceedings of the NATO-ASI meeting, Harmonic Analysis 2000-A Celebration, Kluwer Publisher, to appear in 2001.
- Zimmer, R.J.Ergodic Theory and Semisimple Groups, Birkhäuser, Boston, (1980).
- A characterization of the higher dimensional groups associated with continuous wavelets
The Journal of Geometric Analysis
Volume 12, Issue 1 , pp 89-102
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- continuous wavelets
- discrete wavelets
- admissable dialatin groups