A Simple Remark On The Order Of Approximation
By Compactly Supported Radial Basis Functions,
And Related Networks
We consider simultaneous approximation of multivariate functions and their
derivatives by using Wendland's compactly supported radial basis functions
s,k' By applying a greedy algorithm, it is presented that, regardless of
dimension, an O(ra~*/^) order of approximation can be achieved by a linear
combination of m translates of s,k- A similar result on approximation by
neural networks is established by using univariate radial functions as the
activation functions of the networks.
Multivariate interpolation by radial basis functions has been studied and
applied in several areas of mathematics, such as approximation theory (cf.
Franke^, Micchelli**, Schaback**), curve and surface fitting (cf. Daehlen,
Lyche, and Schumaker*), and numerical soluti