Dunkl harmonic analysis and fundamental sets of continuous functions on the unit sphere

TL;DR

This paper uses Dunkl harmonic analysis to determine when certain constructed functions form a fundamental set in the space of continuous functions on the unit sphere.

Contribution

It provides a necessary and sufficient condition for a continuous function on [-1,1] to generate a fundamental family on the sphere using Dunkl harmonic analysis.

Findings

Established a criterion for fundamentality of function families on the sphere.

Utilized Dunkl harmonic analysis in the construction and proof.

Connected function properties on [-1,1] to spherical function spaces.

Abstract

We establish a necessary and sufficient condition on a continuous function on $[-1,1]$ under which the family of functions on the unit sphere $\mathbb{S}^{d-1}$ constructed in the described manner is fundamental in $C(\mathbb{S}^{d-1})$. In our construction of functions and proof of the result, we essentially use Dunkl harmonic analysis.