On a Choice of Sampling Nodes for Optimal Approximation of Smooth Functions by Generalized Translation Networks

We describe a system of nodes in [-1, 1]^s, where s≥1 is an integer, with the following property. Suppose the values at these nodes of any target function f having a prescribed number r of continuous derivatives on [-1, 1]^s are known. Given any ε>0, we can construct a generalized translation network with O(ε^-s/r) number of principal elements that approximates f uniformly within ε. Within a constant multiple, this is the theoretically minimal number of principal elements required for this purpose. The system of nodes is independent of the target function. We also investigate the effect of small perturbations in this system of nodes on the degree of approximation.