Fundamental sets of continuous functions on spheres. (English) Zbl 0886.41016

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.


41A30 Approximation by other special function classes
41A45 Approximation by arbitrary linear expressions
41A63 Multidimensional problems
42A65 Completeness of sets of functions in one variable harmonic analysis
42A82 Positive definite functions in one variable harmonic analysis
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
42C30 Completeness of sets of functions in nontrigonometric harmonic analysis
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