Dunkl harmonic analysis and fundamental sets of continuous functions on   the unit sphere
  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. 
Roman Veprintsev
math.CA