Approximation properties of zonal function networks using scattered data on the sphere

A zonal function (ZF) network is a function of the form x↦∑ _k=1 ⁿ c _k (x · y _k), where x and the y _k's are on the unit sphere in q+1 dimensional Euclidean space, and where the y _k's are scattered points. In this paper, we study the degree of approximation by ZF networks. In particular, we compare this degree of approximation with that obtained with the classical spherical harmonics. In many cases of interest, this is the best possible for a given amount of information regarding the target function. We also discuss the construction of ZF networks using scattered data. Our networks require no training in the traditional sense, and provide theoretically predictable rates of approximation. This revised version was published online in June 2006 with corrections to the Cover Date.