Wavelet transform and orthogonal decomposition of space $L^2$ on the Cartan domain $BDI(q=2)$
HTML articles powered by AMS MathViewer
- by Qingtang Jiang PDF
- Trans. Amer. Math. Soc. 349 (1997), 2049-2068 Request permission
Abstract:
Let $G=\left ({\mathbb {R}}^{*}_{+}\times SO_{0}(1, n)\right ) \ltimes {\mathbb {R}}^{n+1}$ be the Weyl-Poincar$\acute e$ group and $KAN$ be the Iwasawa decomposition of $SO_{0}(1, n)$ with $K=SO(n)$. Then the “affine Weyl-Poincar$\acute e$ group” $G_{a}=\left ({\mathbb {R}}^{*}_{+}\times AN\right ) \ltimes {\mathbb {R}}^{n+1}$ can be realized as the complex tube domain $\Pi ={\mathbb {R}}^{n+1}+iC$ or the classical Cartan domain $BDI(q=2)$. The square-integrable representations of $G$ and $G_{a}$ give the admissible wavelets and wavelet transforms. An orthogonal basis $\{ \psi _{k}\}$ of the set of admissible wavelets associated to $G_{a}$ is constructed, and it gives an orthogonal decomposition of $L^{2}$ space on $\Pi$ (or the Cartan domain $BDI(q=2)$) with every component $A_{k}$ being the range of wavelet transforms of functions in $H^{2}$ with $\psi _{k}$.References
- 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). MR 1122655
- 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. MR 0393335, DOI 10.1016/0022-1236(76)90079-3
- J. Faraut and A. Korányi, Analysis on symmetric cones, Clarendon Press, Oxford, 1994.
- Hans G. Feichtinger and K. H. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. I, J. Funct. Anal. 86 (1989), no. 2, 307–340. MR 1021139, DOI 10.1016/0022-1236(89)90055-4
- A. Grossmann and J. Morlet, Decomposition of Hardy functions into square integrable wavelets of constant shape, SIAM J. Math. Anal. 15 (1984), no. 4, 723–736. MR 747432, DOI 10.1137/0515056
- L. K. Hua, Harmonic analysis of functions of several complex variables in the classical domains, American Mathematical Society, Providence, R.I., 1963. Translated from the Russian by Leo Ebner and Adam Korányi. MR 0171936, DOI 10.1090/mmono/006
- Q. Jiang, Wavelet transform associated to the Weyl-Poincar$\acute e$ group, preprint.
- Qing Tang Jiang and Li Zhong Peng, Phase space, wavelet transform and Toeplitz-Hankel type operators, Israel J. Math. 89 (1995), no. 1-3, 157–171. MR 1324459, DOI 10.1007/BF02808198
- A. Perelomov, Generalized coherent states and their applications, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1986. MR 858831, DOI 10.1007/978-3-642-61629-7
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
- A. R. Prasanna, A new invariant for electromagnetic fields in curved space-time, Phys. Lett. 37A (1971), 331–332. MR 309522, DOI 10.1016/0375-9601(71)90694-3
- Michael E. Taylor, Noncommutative harmonic analysis, Mathematical Surveys and Monographs, vol. 22, American Mathematical Society, Providence, RI, 1986. MR 852988, DOI 10.1090/surv/022
- André Unterberger and Julianne Unterberger, A quantization of the Cartan domain $BD\textrm {I}\;(q=2)$ and operators on the light cone, J. Funct. Anal. 72 (1987), no. 2, 279–319. MR 886815, DOI 10.1016/0022-1236(87)90090-5
- André Unterberger, Analyse harmonique et analyse pseudo-différentielle du cône de lumière, Astérisque 156 (1987), 201 pp. (1988) (French, with English summary). MR 947371
- Stephen Vági, Harmonic analysis on Cartan and Siegel domains, Studies in harmonic analysis (Proc. Conf., DePaul Univ., Chicago, Ill., 1974) MAA Stud. Math., Vol. 13, Math. Assoc. Amer., Washington, D.C., 1976, pp. 257–309. MR 0477178
- N. Ja. Vilenkin, Special functions and the theory of group representations, Translations of Mathematical Monographs, Vol. 22, American Mathematical Society, Providence, R.I., 1968. Translated from the Russian by V. N. Singh. MR 0229863, DOI 10.1090/mmono/022
Additional Information
- Qingtang Jiang
- Affiliation: Department of Mathematics, Peking University, Beijing 100871, P. R. China
- Address at time of publication: Department of Mathematics, The National University of Singapore, Lower Kent Ridge Road, Singapore 119260
- Email: qjiang@haar.math.nus.sg
- Received by editor(s): November 20, 1994
- Received by editor(s) in revised form: December 2, 1995
- © Copyright 1997 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 349 (1997), 2049-2068
- MSC (1991): Primary 22D10, 81R30; Secondary 42C99
- DOI: https://doi.org/10.1090/S0002-9947-97-01727-3
- MathSciNet review: 1373641