L^p Bernstein Inequalities and Radial Basis Function Approximation

摘要

In approximation theory, three classical types of results are direct theorems,Bernstein inequalities, and inverse theorems. In this paper, we include results aboutradial basis function (RBF) approximation from all three classes. Bernstein inequalitiesare a recent development in the theory of RBF approximation, and on Rd, onlyL2 results are known for RBFs with algebraically decaying Fourier transforms (e.g.the Sobolev splines and thin-plate splines). We will therefore extend what is knownby establishing Lp Bernstein inequalities for RBF networks on Rd. These inequalitiesinvolve bounding a Bessel-potential norm of an RBF network by its corresponding Lpnorm in terms of the separation radius associated with the network. While Bernsteininequalities have a variety of applications in approximation theory, they are most commonlyused to prove inverse theorems. Therefore, using the Lp Bernstein inequalitiesfor RBF approximants, we will establish the corresponding inverse theorems. Thedirect theorems of this paper relate to approximation in Lp(Rd) by RBFs which areperturbations of Green's functions. Results of this type are known for certain compactdomains, and results have recently been derived for approximation in Lp(Rd)by RBFs that are Green's functions. Therefore, we will prove that known results forapproximation in Lp(Rd) hold for a larger class of RBFs. We will then show how thisresult can be used to derive rates for approximation by Wendland functions.