The authors study the approximation of a function on the surface of the unit sphere in the Euclidean space of dimension
. They define a zonal function network to be a finite linear combination of functions
. 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
under certain minimal conditions. For certain natural function classes, to which the target functions are assumed to belong to and
satisfying additional conditions, the results obtained are close to optimal.