Nonlocal Operational Calculi for Dunkl Operators

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.