Fundamental sets of continuous functions on spheres (Q1357859)

Let \(S^m\) and \(S^\infty\) denote the unit spheres in \(\mathbb{R}^{m+1}\) and \(\ell^2\), respectively. The authors look for functions \(f\) in \(C[- 1,1]\) such that the family of functions \(x\mapsto f(\langle x,v\rangle)\), where \(v\) runs over \(S^m\), is fundamental in the space \(C(S^m)\). They also consider this problem for \(C(S^\infty)\) when this space is given the topology of uniform convergence on compact sets.