The Continuous Wavelet Transform Associated with a Dunkl Type Operator on the Real Line


We consider a singular differential-difference operator Λ on R which includes as a particular case the one-dimensional Dunkl operator. By using harmonic analysis tools corresponding to Λ, we introduce and study a new continuous wavelet transform on R tied to Λ. Such a wavelet transform is exploited to invert an intertwining operator between Λ and the first derivative operator d/dx.

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E. Zahrani and M. Mourou, "The Continuous Wavelet Transform Associated with a Dunkl Type Operator on the Real Line," Advances in Pure Mathematics, Vol. 3 No. 5, 2013, pp. 443-450. doi: 10.4236/apm.2013.35063.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] C. F. Dunkl, “Differential-Difference Operators Associated to Refection Groups,” Transactions of the American Mathematical Society, Vol. 311, No. 1, 1989, pp. 167-183. doi:10.1090/S0002-9947-1989-0951883-8
[2] C. F. Dunkl, “Integral Kernels with Reflection Group Invariance,” Canadian Journal of Mathematics, Vol. 43, No. 6, 1991, pp. 1213-1227. doi:10.4153/CJM-1991-069-8
[3] C. F. Dunkl, “Hankel Transforms Associated to Finite Reflection Groups,” Contemporary Mathematics, Vol. 138, No. 1, 1992, pp. 128-138.
[4] S. Kamefuchi and Y. Ohnuki, “Quantum Field Theory and Parastatistics,” University of Tokyo Press, Springer-Verlag, Berlin, 1982.
[5] M. Rosenblum, “Generalized Hermite Polynomials and the Bose like Oscillator Calculus,” In: Operator Theory: Advances and Applications, Birkhauser Verlag, 1994, pp. 369-396.
[6] L. M. Yang, “A Note on the Quantum Rule of the Harmonic Oscillator,” Physical Review, Vol. 84, No. 4, 1951, pp. 788-790. doi:10.1103/PhysRev.84.788
[7] E. A. Al Zahrani, H. El Mir and M. A. Mourou, “Intertwining Operators Associated with a Dunkl Type Operator on the Real Line and Applications,” Far East Journal of Applied Mathematics and Applications, Vol. 64, No. 2, 2012, pp. 129-144.
[8] A. Grossmann and J. Morlet, “Decomposition of Hardy Functions Intosquare Integrable Wavelets of Constant Shape,” SIAM Journal on Mathematical Analysis, Vol. 15, No. 4, 1984, pp. 723-736. doi:10.1137/0515056
[9] C. K. Chui, “An Introduction to Wavelets,” Academic Press, Waltham, 1992.
[10] T. H. Koornwinder, “The Continuous Wavelet Transform,” In: T. H. Koornwinder, Ed., Wavelets: An Elementary Treatment of Theory and Applications, World Scientific, Singapore City, 1993, pp. 27-48.
[11] Y. Meyer, “Wavelets and Operators,” Cambridge University Press, Cambridge, 1992.
[12] I. Daubechies, “Ten Lectures on Wavelets,” CBMS-NSF Regional Conference Series in Applied Mathematics, Philadelphia, 1992.
[13] P. Goupillaud, A. Grossmann and J. Morlet, “Cycle Oc-Tave and Related Transforms in Seismic Signal Analysis,” Geoexploration, Vol. 23, No. 1, 1984, pp. 85-102. doi:10.1016/0016-7142(84)90025-5
[14] M. Holschneider, “Wavelets: An Analysis Tool,” Clarendon Press, Oxford, 1995.
[15] M. Rosler, “Bessel-Type Signed Hypergroups on R,” In: H. Heyer and A. Mukherjea, Eds., Probability Measures on Groups and Related Structures World Scientific, Singapore City, 1995, pp. 292-304.
[16] M. A. Mourou and K. Trimeche, “Calderon’s Reproducing Formula Related to the Dunkl Operator on the Real Line,” Monatshe fur Mathematik, Vol. 136, No. 1, 2002, 47-65. doi:10.1007/s006050200033
[17] M. A. Mourou, “Inversion of the Dual Dunkl-Sonine Transform on R Using Dunkl Wavelets,” SIGMA, Vol. 5, No. 71, 2009, pp. 1-12.

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