Extending error bounds for radial basis function interpolation to measuring the error in higher order Sobolev norms (Q6622394)

The article provides a new family of Bernstein inequalities and applies them to remarkable radial basis function (RBF) classes, including surface splines, Matérn kernels, and compactly supported kernels, such as Wendland's minimal-degree kernels. Additionally, it introduces an integral-based approximation scheme, analyzes its error, and proves its utility across these RBF families. This analysis provides sharp interpolation error bounds, both for positive definite RBFs, such as certain Matérn and compactly supported kernels, and for conditionally positive definite RBFs, including those with Fourier transforms that have algebraic singularities. New convergence rates are obtained for the abstract operator studied by Devore and Ron, along with high-order Bernstein estimates linking smoothness norms to the native space norm, contributing to a deeper understanding of RBF interpolation in settings that require enhanced smoothness.