Generalized Whittle-Matern and polyharmonic kernels

This paper simultaneously generalizes two standard classes of radial kernels, the polyharmonic kernels related to the differential operator ( - Δ) m and the Whittle-Matérn kernels related to the differential operator ( - Δ + I) m. This is done by allowing general differential operators of the form {Mathematical expression} with nonzero κ j and calculating their associated kernels. It turns out that they can be explicity given by starting from scaled Whittle-Matérn kernels and taking divided differences with respect to their scale. They are positive definite radial kernels which are reproducing kernels in Hilbert spaces norm-equivalent to {Mathematical expression}. On the side, we prove that generalized inverse multiquadric kernels of the form {Mathematical expression} are positive definite, and we provide their Fourier transforms. Surprisingly, these Fourier transforms lead to kernels of Whittle-Matérn form with a variable scale κ(r) between κ 1,..., κ m. We also consider the case where some of the κ j vanish. This leads to conditionally positive definite kernels that are linear combinations of the above variable-scale Whittle-Matérn kernels and polyharmonic kernels. Some numerical examples are added for illustration