Generalized Whittle-Mat,rn and polyharmonic kernels

This paper simultaneously generalizes two standard classes of radial kernels, the polyharmonic kernels related to the differential operator ( -aEuro parts per thousand Delta) (m) and the Whittle-Mat,rn kernels related to the differential operator ( -aEuro parts per thousand Delta + I) (m) . This is done by allowing general differential operators of the form with nonzero kappa (j) and calculating their associated kernels. It turns out that they can be explicity given by starting from scaled Whittle-Mat,rn 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 . On the side, we prove that generalized inverse multiquadric kernels of the form are positive definite, and we provide their Fourier transforms. Surprisingly, these Fourier transforms lead to kernels of Whittle-Mat,rn form with a variable scale kappa(r) between kappa (1),...,kappa (m) . We also consider the case where some of the kappa (j) vanish. This leads to conditionally positive definite kernels that are linear combinations of the above variable-scale Whittle-Mat,rn kernels and polyharmonic kernels. Some numerical examples are added for illustration.