Research ArticleOpen AccessGoogle Scholar indexed
The Commutants of the Dunkl Operators on <i> ε</i>(R<sup>d</sup>)
Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis, Tunis, Tunisia
Higher College of Technology and Informatics, Tunis, Tunisia
- 1 Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis, Tunis, Tunisia
- 2 Higher College of Technology and Informatics, Tunis, Tunisia
Advances in Pure Mathematics·Volume 02 (2012)·Pages 274–297·Published 30 July 2012·DOI10.4236/apm.2012.24035
Copy link · social · email
Abstract
We consider the harmonic analysis associated with the Dunkl operators on R d . We study the Dunkl mean-periodic functions on the space ε (R d ) (the space of C ∞ -functions). We characterize also the continuous linear mappings from ε (R d ) into itself which commute with the Dunkl operators.
KeywordsDunkl Operators on R<sup>d</sup>C<sup>∞</sup>-Functions on R<sup>d</sup>Dunkl Intertwining OperatorMean-Periodic FunctionsContinuous Linear Mappings
- 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
- 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
- M. F. E. de Jeu, “The Dunkl Transform,” Inventiones Mathematicae, Vol. 113, No. 1, 1993, pp. 147-162. doi:10.1007/BF01244305
- 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
- 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
- M. R?sler, “Positivity of Dunkl’s Intertwining Operator,” Duke Mathematical Journal, Vol. 98, 1999, pp. 445-463.
- 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
- 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.
- 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
- L. H?rmander, “Linear Partial Differential Operators,” Springer-Verlag, Berlin, Gottingen, Heidelberg, 1964.
- L. Schwartz, “Théorie Générale des Fonctions Moyennes Périodiques,” Annals of Mathematics, Vol. 48, No. 4, 1947, pp. 857-929. doi:10.2307/1969386