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

Affiliation(s)

Department of Mathematics, College of Sciences for Girls, University of Dammam, Dammam, KSA.

Department of Mathematics, College of Sciences for Girls, University of Dammam, Dammam, KSA.

ABSTRACT

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/d*x*.

KEYWORDS

Differential-Difference Operator; Generalized Wavelets; Generalized Continuous Wavelet Transform

Differential-Difference Operator; Generalized Wavelets; Generalized Continuous Wavelet Transform

Cite this paper

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.

E. Zahrani and M. Mourou, "The Continuous Wavelet Transform Associated with a Dunkl Type Operator on the Real Line,"

References

[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.

[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.