The essential rate of approximation for radial function manifold

In this paper, we investigate the radial function manifolds generated by a linear combination of radial functions. Let W_p^rbe the usual Sobolev class of functions on the unit ball. We study the deviation from the radial function manifolds to W_p^r.Our results show that the upper and lower bounds of approximation by a linear combination of radial functions are asymptotically identical. We also find that the radial function manifolds and ridge function manifolds generated by a linear combination of ridge functions possess the same rate of approximation.