$L_p$-error estimates for ``shifted'' surface spline interpolation on Sobolev space

The accuracy of interpolation by a radial basis function $\phi$ is usually very satisfactory provided that the approximant $f$ is reasonably smooth. However, for functions which have smoothness below a certain order associated with the basis function $\phi$, no approximation power has yet been established. Hence, the purpose of this study is to discuss the $L_p$-approximation order ( $1\leq p\leq \infty$) of interpolation to functions in the Sobolev space $W^k_p(\Omega)$ with $k> \max(0,d/2-d/p)$. We are particularly interested in using the ``shifted'' surface spline, which actually includes the cases of the multiquadric and the surface spline. Moreover, we show that the accuracy of the interpolation method can be at least doubled when additional smoothness requirements and boundary conditions are met.