Nonlocal Operational Calculi for Dunkl Operators

TL;DR

This paper develops an operational calculus for Dunkl operators with nonlocal boundary conditions, enabling solutions to nonlocal differential equations and establishing conditions for unique solutions in mean-periodic functions.

Contribution

It introduces a Mikusinski-type operational calculus for Dunkl operators and extends the Heaviside algorithm to solve nonlocal boundary value problems.

Findings

Constructed a right inverse operator for Dunkl operator under nonlocal conditions

Extended the Heaviside algorithm for Dunkl functional-differential equations

Derived conditions for the existence and uniqueness of solutions in mean-periodic functions

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.