Best approximation to C² functions and its error bounds using Gaussian hidden units

In the present paper we prove that any C² function of m real variables with support in the unit hypercube can be approximated by a Gaussian radial basis network. This network uses a single layer of N Gaussian radial basis functions. The centers of the Gaussian functions are uniformly distributed on the unit hypercube. From the view point of the best approximation theory, an upper bound of this approximation O(σ²+N^-2) is obtained, where σ is the deviation of Gaussians. Our results provide an explicit expression of the relationship between the number of hidden nodes and the approximation error.