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Fundamental sets of continuous functions on spheres. (English) Zbl 0886.41016
Let $S^m$ and $S^\infty$ denote the unit spheres in $\bbfR^{m+1}$ 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.

MSC:
41A30Approximation by other special function classes
41A45Approximation by arbitrary linear expressions
41A63Multidimensional approximation problems
42A65Completeness of sets of functions
42A82Positive definite functions
42C10Fourier series in special orthogonal functions
42C30Completeness of sets of functions of non-trigonometric Fourier analysis
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References:
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