SIGMA 5 (2009), 030, 16 pages      arXiv:0903.1609      https://doi.org/10.3842/SIGMA.2009.030
Contribution to the Special Issue on Dunkl Operators and Related Topics

Nonlocal Operational Calculi for Dunkl Operators

Ivan H. Dimovski and Valentin Z. Hristov
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 8, 1113 Sofia, Bulgaria

Received October 15, 2008, in final form March 04, 2009; Published online March 09, 2009

Abstract
The one-dimensional Dunkl operator D_k with a non-negative parameter k, is considered under an arbitrary nonlocal boundary value condition. The right inverse operator of D_k, satisfying this condition is studied. An operational calculus of Mikusinski type is developed. In the frames of this operational calculi an extension of the Heaviside algorithm for solution of nonlocal Cauchy boundary value problems for Dunkl functional-differential equations P(D_k)u = f with a given polynomial P is proposed. The solution of these equations in mean-periodic functions reduces to such problems. Necessary and sufficient condition for existence of unique solution in mean-periodic functions is found.

Key words: Dunkl operator; right inverse operator; Dunkl-Appell polynomials; convolution; multiplier; multiplier fraction; Dunkl equation; nonlocal Cauchy problem; Heaviside algorithm; mean-periodic function.

pdf (273 kb)   ps (182 kb)   tex (16 kb)

References

1. Betancor J.J., Sifi M., Trimèche K., Intertwining operator and the commutators of the Dunkl operator on C, Math. Sci. Res. J. 10 (2006), no. 3, 66-78.
2. Bittner R., Operational calculus in linear spaces, Studia Math. 20 (1961), 1-18.
3. Ben Salem N., Kallel S., Mean-periodic functions associated with the Dunkl operators, Integral Transforms Spec. Funct. 15 (2004), 155-179.
4. Delsarte J., Les fonctions moyenne-périodiques, J. Math. Pures Appl. 14 (1935), 403-453.
5. Dimovski I.H., Convolutional calculus, Kluwer Academic Publishers Group, Dordrecht, 1990.
6. Dimovski I.H., Nonlocal operational calculi, Proc. Steklov Inst. Math. 203 (1995), no. 3, 53-65.
7. Dimovski I.H., Hristov V.Z., Sifi M., Commutants of the Dunkl operators in C(R), Fract. Calc. Appl. Anal. 9 (2006), 195-213.
8. Dunkl C.F., Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), 167-183.
9. Larsen R., An introduction to the theory of multipliers, Springer-Verlag, New York - Heidelberg, 1971.
10. Mikusinski J., Operational calculus, I, Warszawa, 1967.
11. Mourou M.A., Trimèche K., Opérateurs de transmutation et théorème de Paley-Wiener associés à un opérateur aux dérivées et différences sur R, C. R. Acad. Sci. Paris Ser. I Math. 332 (2001), 397-400.
12. Przeworska-Rolewicz D., Algebraic theory of right inverse operators, Studia Math. 48 (1973), 129-144.
13. Rösler M., Bessel-type signed hypergroups on R, in Probability Measures on Groups and Related Structures XI (Oberwolfach, 1994), World Sci. Publ., River Edge, NJ, 1995, 292-304.
14. Rösler M., Voit M., Biorthogonal polynomials associated with reflection groups and a formula of Macdonald, J. Comput. Appl. Math. 99 (1998), 337-351, q-alg/9711004.
15. Trimèche K., The Dunkl intertwining operator on spaces of functions and distributions and integral representation of its dual, Integral Transforms Spec. Funct. 12 (2001), 349-374.
16. Xu Y., An integral formula for generalized Gegenbauer polynomials and Jacobi polynomials, Adv. in Appl. Math. 29 (2002), 328-341.