We consider a singular differential-difference operator $\Lambda$ on $\R$ which generalizes the one-dimensional Dunkl operator. Using harmonic analysis associated with $\Lambda$ we define and study three types of generalized wavelet packets and the corresponding wavelet transforms. As an application, we introduce on $\R$ a new multiresolution analysis tied to $\Lambda$.
References
W. Chabeh and M. A. Mourou, Inversion of the generalized Dunkl intertwining operator on $\R$ and its dual using generalized wavelets. Eur. J. Pure Appl. Math. 3 (2010), pp 958-979. MR2749544 1211.42030 W. Chabeh and M. A. Mourou, Inversion of the generalized Dunkl intertwining operator on $\R$ and its dual using generalized wavelets. Eur. J. Pure Appl. Math. 3 (2010), pp 958-979. MR2749544 1211.42030
C. K. Chui, An introduction to wavelets. Academic Press, New York (1992). MR1150048 0925.42016 C. K. Chui, An introduction to wavelets. Academic Press, New York (1992). MR1150048 0925.42016
R. R. Coifman and M.V. Wickerhauser, Entropy-Based Algorithms for Best Basis Selection. IEEE Transactions on Information Theory 38 (1992), pp 1241-1243. R. R. Coifman and M.V. Wickerhauser, Entropy-Based Algorithms for Best Basis Selection. IEEE Transactions on Information Theory 38 (1992), pp 1241-1243.
C. F. Dunkl, Integral kernels with reflection group invariance. Can. J. Math. 43 (1991), pp 1213-1227. MR1145585 10.4153/CJM-1991-069-8 0827.33010 C. F. Dunkl, Integral kernels with reflection group invariance. Can. J. Math. 43 (1991), pp 1213-1227. MR1145585 10.4153/CJM-1991-069-8 0827.33010
C. F. Dunkl, Hankel transforms associated to finite reflection groups. Contemp. Math. 138 (1992), pp 128-138. MR1199124 0789.33008 10.1090/conm/138/1199124 C. F. Dunkl, Hankel transforms associated to finite reflection groups. Contemp. Math. 138 (1992), pp 128-138. MR1199124 0789.33008 10.1090/conm/138/1199124
M. Duval-Destin, M.N. Muschietti and M. Torresani, Continuous wavelet decomposition, multiresolution and contrast analysis. SIAM J. Math. Anal. 24 (1993), pp 739-755. MR1215435 0770.41023 10.1137/0524045 M. Duval-Destin, M.N. Muschietti and M. Torresani, Continuous wavelet decomposition, multiresolution and contrast analysis. SIAM J. Math. Anal. 24 (1993), pp 739-755. MR1215435 0770.41023 10.1137/0524045
W. Freeden and U. Windheuser, Spherical wavelet transform and its discretization. Advances in comput. Math. 5 (1996), pp 51-94. MR1381282 0858.44005 10.1007/BF02124735 W. Freeden and U. Windheuser, Spherical wavelet transform and its discretization. Advances in comput. Math. 5 (1996), pp 51-94. MR1381282 0858.44005 10.1007/BF02124735
T. H. Koornwinder, The continuous wavelet transform. Wavelets : An Elementary Treatement of Theory and Applications. Edited by T. H. Koornwinder, World Scientific 1 (1993), pp 27-48. MR1238554 T. H. Koornwinder, The continuous wavelet transform. Wavelets : An Elementary Treatement of Theory and Applications. Edited by T. H. Koornwinder, World Scientific 1 (1993), pp 27-48. MR1238554
L. Lapointe and L. Vinet, Exact operator solution of the Calogero-Sutherland model. Comm. Math. Phys. 178 (1996), pp 425-452. MR1389912 0859.35103 10.1007/BF02099456 euclid.cmp/1104286659
L. Lapointe and L. Vinet, Exact operator solution of the Calogero-Sutherland model. Comm. Math. Phys. 178 (1996), pp 425-452. MR1389912 0859.35103 10.1007/BF02099456 euclid.cmp/1104286659
M. A. Mourou and K. Trimeche, Transmutation operators and Paley-Wiener theorem associated with a singular differential-difference operator on the real line. Anal. Appl. 1 (2003), pp 43-69. MR1959284 1140.42302 10.1142/S0219530503000090 M. A. Mourou and K. Trimeche, Transmutation operators and Paley-Wiener theorem associated with a singular differential-difference operator on the real line. Anal. Appl. 1 (2003), pp 43-69. MR1959284 1140.42302 10.1142/S0219530503000090
