Strictly Positive Definite Functions on Compact Two-Point Homogeneous Spaces: the Product Alternative

For two continuous and isotropic positive definite kernels on the same compact two-point homogeneous space, we determine necessary and sufficient conditions in order that their product be strictly positive definite. We also provide a similar characterization for kernels on the space-time setting \(G \times S^d\), where \(G\) is a locally compact group and \(S^d\) is the unit sphere in \(\mathbb{R}^{d+1}\), keeping isotropy of the kernels with respect to the \(S^d\) component. Among other things, these results provide new procedures for the construction of valid models for interpolation and approximation on compact two-point homogeneous spaces.