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Nonlocal operational calculi for Dunkl operators. (English) Zbl 1163.44003

Summary: 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 the 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.

MSC:

44A40 Calculus of Mikusiński and other operational calculi
44A35 Convolution as an integral transform
34K06 Linear functional-differential equations
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
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