  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.
