Fundamental sets of continuous functions on spheres
Let Sm and S∞ denote the unit spheres in Rm+1 and ℓ2, respectively. We look for functions f in C[-1, 1] such that the family of functions x → f (〈x, v〉) is fundamental in the space C(Sm). Here v runs over Sm. There is a similar question for C(S∞), when this space is given the topology of uniform convergence on compact sets.
Approximation, Continuous functions, Fundamentally, Positive-definite, Spheres
Sun, Xingping, and E. W. Cheney. "Fundamental sets of continuous functions on spheres." Constructive Approximation 13, no. 2 (1997): 245-250.