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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 Dk with a non-negative parameter
k, is considered under an arbitrary nonlocal boundary value
condition. The right inverse operator of Dk, 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(Dk)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.
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