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

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.