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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 theresult, we essentially use Dunkl harmonic analysis.

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