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

Abstract

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×Sᵈ, where G is a locally compact group, and Sᵈ is the unit sphere in ℝᵈ⁺¹, keeping isotropy of the kernels with respect to the Sᵈ 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.