Dunkl positive definite functions

TL;DR

This paper introduces Dunkl positive definite functions, extends Bochner's theorem to the Dunkl setting, and provides conditions for strict positive definiteness, advancing harmonic analysis in this generalized framework.

Contribution

It defines Dunkl positive definite functions, establishes an analogue of Bochner's theorem, and offers criteria for strict positive definiteness in the Dunkl context.

Findings

Dunkl positive definite functions are characterized using Dunkl translation.

An analogue of Bochner's theorem is proven for Dunkl setting.

Sufficient conditions for strict positive definiteness are provided.

Abstract

We introduce the notion of Dunkl positive definite and strictly positive definite functions on $\mathbb{R}^{d}$. This done by the use of the properties of Dunkl translation. We establish the analogue of Bochner's theorem in Dunkl setting. The case of radial functions is considered. We give a sufficient condition for a function to be Dunkl strictly positive definite on $\mathbb{R}^{d}.$