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 x S-d, where G is a locally compact group and S-d is the unit sphere in Rd+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.