Differentiable positive definite functions on two-point homogeneous spaces

TL;DR

This paper extends the understanding of positive definite kernels on compact two-point homogeneous spaces, generalizing known differentiability properties from spheres to other such spaces.

Contribution

It provides a generalization of differentiability results for positive definite kernels from spheres to all compact two-point homogeneous spaces.

Findings

Differentiability properties of kernels are established for various spaces.

The results generalize previous sphere-specific findings.

Implications for scattered data interpolation are discussed.

Abstract

In this paper we study continuous kernels on compact two point homogeneous spaces which are positive definite and zonal (isotropic). Such kernels were characterized by R. Gangolli some forty years ago and are very useful for solving scattered data interpolation problems on the spaces. In the case the space is the $d$-dimensional unit sphere, J. Ziegel showed in 2013 that the radial part of a continuous positive definite and zonal kernel is continuously differentiable up to order $\lfloor (d-1)/2 \rfloor$ in the interior of its domain. The main issue here is to obtain a similar result for all the other compact two point homogeneous spaces.