Spectral Approximation Orders of Radial Basis Function Interpolation on the Sobolev Space

Abstract

In this study, we are mainly interested in error estimates of interpolation, using smooth radial basis functions such as multiquadrics. The current theories of radial basis function interpolation provide optimal error bounds when the basis function $\phi$ is smooth and the approximand f is in a certain reproducing kernel Hilbert space ${\mathcal F}_\phi$. However, since the space ${\mathcal F}_\phi$ is very small when the function $\phi$ is smooth, the major concern of this paper is to prove approximation orders of interpolation to functions in the Sobolev space. For instance, when $\phi$ is a multiquadric, we will observe the error bound $o(h^k)$ if the function to be approximated is in the Sobolev space of smoothness order k.