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Approximation properties of zonal function networks using scattered data on the sphere. (English) Zbl 0939.41012
The authors study the approximation of a function on the surface of the unit sphere in the Euclidean space of dimension $\left(q+1\right)$, $q\ge 1$. They define a zonal function network to be a finite linear combination of functions $x\to \phi \left(x,{y}_{k}\right)$. They compare the degree of approximation by zonal function networks with the degree of approximation provided by spherical harmonics. They obtain general results valid for essentially arbitrary target functions and all $\phi$ under certain minimal conditions. For certain natural function classes, to which the target functions are assumed to belong to and $\phi$ satisfying additional conditions, the results obtained are close to optimal.

##### MSC:
 41A30 Approximation by other special function classes 30C10 Polynomials (one complex variable)