Dunkl regularity over alternative $*$-algebras

TL;DR

This paper introduces Dunkl-regular functions over real alternative *-algebras, unifying hypercomplex analysis with Dunkl operator theory and expanding the framework of Dunkl monogenic functions.

Contribution

It characterizes slice-regularity using Dunkl operators and defines new function spaces that generalize existing hypercomplex function theories.

Findings

Dunkl-regular functions refine Dunkl monogenic and harmonic analysis.

A unifying framework for various hypercomplex function theories.

Potential for new interactions between Dunkl theory and hypercomplex analysis.

Abstract

We characterise slice-regularity of functions over a real alternative *-algebra using operators that arise in Dunkl operator theory. We present a unifying perspective on hypercomplex analysis by defining a family of function spaces in the kernel of Dunkl-Cauchy-Riemann operators. Each of these function spaces, whose elements are called Dunkl-regular functions, refines Dunkl monogenic function theory and Dunkl harmonic analysis on Euclidean spaces. This approach allows a wide variety of hypercomplex function theories to be embedded as subcases of Dunkl monogenic function theory. This paves the way for further interactions between Dunkl theory and hypercomplex analysis.