ABSTRACT

Besides using standard radial basis functions for recovery of functions from data or as solutions of PDEs, there are a few good reasons to look for kernels with special properties. This talk will provide several examples, starting from an introduction into kernel construction techniques. After providing the “missing” Wendland kernels, the focus will move to kernels based on series expansions. These have some very interesting special cases, namely polynomial and periodic kernels, and “Taylor” kernels for which the reproduction formula coincides with the Taylor formula. Finally, the use of kernels as particular or fundamental solutions of PDEs is reviewed, together with harmonic kernels and kernels generating divergence–free vector fields. Numerical examples will be provided.