Nonstandard Kernels and their Applications

Besides using standard radial basis functions, there are a few good reasons to look for kernels with special properties. This survey will provide several examples, starting from an introduction into kernel construction techniques. After providing the “missing” Wendland functions, we focus on 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, we review the use of kernels as particular or fundamental solutions of PDEs, look at harmonic kernels and kernels generating divergence–free vector fields. Numerical examples will be provided as we are going along