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

A zonal function (ZF) network is a function of the form x 7→ ∑n k=1 ckφ(x · yk), where x and the yk’s are on the on the unit sphere in q+1 dimensional Euclidean space, and where the yk’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. ∗Research of the authors was sponsored by the Air Force Office of Scientific Research, Air Force Materiel Command, USAF, under grant numbers F49620-97-1-0211 and F4962098-1-0204. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Office of Scientific Research or the U.S. Government.