Admissible vectors for the regular representation. (English) Zbl 1002.43004
It is 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. The author discusses the case 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 proved by the author states that this restriction disappears in the nonunimodular case. For a nondiscrete second countable group \(G\) with type \(I\) regular representation \(\lambda_G\), it is shown that \(\lambda_G\) itself, and hence every subrepresentation thereof, has an admissible vector in the sense of wavelet theory iff \(G\) is nonunimodular.
Reviewer: Ram Kishore Saxena (Jodhpur)
MSC:
| 43A30 | Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. |
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
Keywords:
continuous wavelet transforms; coherent states; square-integrable representations; Plancherel theory; cyclic vectors; locally compact group; wavelet transformsReferences:
| [1] | S. T. Ali, J-P. Antoine and J-P. Gazeau, Coherent States, Wavelets and Their Generalizations, Springer-Verlag, New York, 2000. CMP 2000:07 · Zbl 1064.81069 |
| [2] | S.T. Ali, H. Führ and A. Krasowska, Plancherel inversion as unified approach to wavelet transforms and Wigner functions, submitted. · Zbl 1049.81043 |
| [3] | Didier Arnal and Jean Ludwig, Q.U.P. and Paley-Wiener properties of unimodular, especially nilpotent, Lie groups, Proc. Amer. Math. Soc. 125 (1997), no. 4, 1071 – 1080. · Zbl 0866.43002 |
| [4] | David Bernier and Keith F. Taylor, Wavelets from square-integrable representations, SIAM J. Math. Anal. 27 (1996), no. 2, 594 – 608. · Zbl 0843.42018 · doi:10.1137/S0036141093256265 |
| [5] | G. Bohnke, Treillis d’ondelettes associés aux groupes de Lorentz, Ann. Inst. H. Poincaré Phys. Théor. 54 (1991), no. 3, 245 – 259 (French, with English summary). · Zbl 0743.43005 |
| [6] | A. L. Carey, Group representations in reproducing kernel Hilbert spaces, Rep. Math. Phys. 14 (1978), no. 2, 247 – 259. · Zbl 0418.22004 · doi:10.1016/0034-4877(78)90047-2 |
| [7] | Jacques Dixmier, \?*-algebras, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett; North-Holland Mathematical Library, Vol. 15. · Zbl 0372.46058 |
| [8] | M. Duflo and Calvin C. Moore, On the regular representation of a nonunimodular locally compact group, J. Functional Analysis 21 (1976), no. 2, 209 – 243. · Zbl 0317.43013 |
| [9] | Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. · Zbl 0857.43001 |
| [10] | Hartmut Führ, Wavelet frames and admissibility in higher dimensions, J. Math. Phys. 37 (1996), no. 12, 6353 – 6366. · Zbl 0864.44003 · doi:10.1063/1.531752 |
| [11] | H. Führ and M. Mayer: Continuous wavelet transforms from semidirect products: Cyclic representations and Plancherel measure. J. Fourier Anal. Appl., to appear. · Zbl 1100.42030 |
| [12] | A. Grossmann, J. Morlet, and T. Paul, Transforms associated to square integrable group representations. I. General results, J. Math. Phys. 26 (1985), no. 10, 2473 – 2479. · Zbl 0571.22021 · doi:10.1063/1.526761 |
| [13] | C. J. Isham and J. R. Klauder, Coherent states for \?-dimensional Euclidean groups \?(\?) and their application, J. Math. Phys. 32 (1991), no. 3, 607 – 620. · Zbl 0735.22012 · doi:10.1063/1.529402 |
| [14] | Qingtang Jiang, Wavelet transform and orthogonal decomposition of \?² space on the Cartan domain \?\?\?(\?=2), Trans. Amer. Math. Soc. 349 (1997), no. 5, 2049 – 2068. · Zbl 0868.22008 |
| [15] | J. R. Klauder and R. F. Streater, A wavelet transform for the Poincaré group, J. Math. Phys. 32 (1991), no. 6, 1609 – 1611. · Zbl 0732.22020 · doi:10.1063/1.529273 |
| [16] | R.S. Laugesen, N. Weaver, G. Weiss and E.N. Wilson, Continuous wavelets associated with a general class of admissible groups and their characterization. J. Geom. Anal., to appear. · Zbl 1043.42032 |
| [17] | Ronald L. Lipsman, Non-Abelian Fourier analysis, Bull. Sci. Math. (2) 98 (1974), no. 4, 209 – 233. · Zbl 0298.43008 |
| [18] | G. Mackey, Borel structure in groups and their duals, Trans. Amer. Math. Soc. 85 (1957), 134-165. · Zbl 0082.11201 |
| [19] | R. Murenzi, Ondelettes multidimensionelles et application à l’analyse d’images, Thèse, Université Catholique de Louvain, Louvain-La-Neuve, 1990. |
| [20] | Nobuhiko Tatsuuma, Plancherel formula for non-unimodular locally compact groups, J. Math. Kyoto Univ. 12 (1972), 179 – 261. · Zbl 0241.22017 |
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