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---Matern kernels related to the differential operator (???Δ?+?I) m . This is done by allowing general differential operators of the form $\prod_{j=1}^m(-\Delta+\kappa_j^2I)$ with nonzero ? j and calculating their associated kernels. It turns out that they can be explicity given by starting from scaled Whittle---Matern 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 $W_2^m(\ensuremath{\mathbb{R}}^d)$ . On the side, we prove that generalized inverse multiquadric kernels of the form $\prod_{j=1}^m(r^2+\kappa_j^2)^{-1}$ are positive definite, and we provide their Fourier transforms. Surprisingly, these Fourier transforms lead to kernels of Whittle---Matern form with a variable scale ?(r) between ? 1,...,? m . We also consider the case where some of the ? j vanish. This leads to conditionally positive definite kernels that are linear combinations of the above variable-scale Whittle---Matern kernels and polyharmonic kernels. Some numerical examples are added for illustration.