The Commutants of the Dunkl Operators on ε(Rd)

Abstract

We consider the harmonic analysis associated with the Dunkl operators on Rd. We study the Dunkl mean-periodic functions on the space ε(Rd) (the space of C-functions). We characterize also the continuous linear mappings from ε(Rd) into itself which commute with the Dunkl operators.

Share and Cite:

M. Sifi and F. Soltani, "The Commutants of the Dunkl Operators on ε(Rd)," Advances in Pure Mathematics, Vol. 2 No. 4, 2012, pp. 274-297. doi: 10.4236/apm.2012.24035.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] C. F. Dunkl, “Differential-Difference Operators Associated with Reflections Groups,” Transactions of the American Mathematical Society, Vol. 311, 1989, pp. 167- 183. doi:10.1090/S0002-9947-1989-0951883-8
[2] C. F. Dunkl, “Integral Kernels with Reflection Group In-variance,” Canadian Journal of Mathematics, Vol. 43, No. 6, 1991, pp. 1213-1227. doi:10.4153/CJM-1991-069-8
[3] M. F. E. de Jeu, “The Dunkl Transform,” Inventiones Mathematicae, Vol. 113, No. 1, 1993, pp. 147-162. doi:10.1007/BF01244305
[4] K. Trimèche, “Paley-Wiener Theorems for the Dunkl Transform and Dunkl Translation Operators,” Integral Transforms and Special Functions, Vol. 13, No. 1, 2002, pp. 17-38. doi:10.1080/10652460212888
[5] K. Trimèche, “The Dunkl Intertwining Operator on Spaces of Functions and Distributions and Integral Representation of Dual,” Integral Transforms and Special Functions, Vol. 12, No. 4, 2001, pp. 349-374. doi:10.1080/10652460108819358
[6] M. R?sler, “Positivity of Dunkl’s Intertwining Operator,” Duke Mathematical Journal, Vol. 98, 1999, pp. 445-463.
[7] J. J. Betancor, M. Sifi and K. Trimèche, “Hypercyclic and Chaotic Convolution Operators Associated with the Dunkl Operator on ,” Acta Mathematica Hungarica, Vol. 106, No. 1-2, 2005, pp. 101-116. doi:10.1007/s10474-005-0009-1
[8] J. J. Betancor, M. Sifi and K. Trimèche, “Intertwining Operator and the Commutators of the Dunkl Operator on ,” Mathematical Sciences Research Journal, Vol. 10, 2006, pp. 66-78.
[9] N. B. Salem and S. Kallel, “Mean-Periodic Functions Associated with the Dunkl Operators,” Integral Transforms and Special Functions, Vol. 15, No. 2, 2004, pp. 155-179. doi:10.1080/10652460310001600735
[10] L. H?rmander, “Linear Partial Differential Operators,” Springer-Verlag, Berlin, Gottingen, Heidelberg, 1964.
[11] L. Schwartz, “Théorie Générale des Fonctions Moyennes Périodiques,” Annals of Mathematics, Vol. 48, No. 4, 1947, pp. 857-929. doi:10.2307/1969386

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.