In this paper, we consider the problem of characterizing positive definite functions on compact two-point homogeneous spaces cross locally compact abelian groups. For a locally compact abelian group
$G$
with dual group
$\widehat{G}$
, a compact two-point homogeneous space
$\mathbb{H}$
with normalized geodesic distance
$\unicode[STIX]{x1D6FF}$
and a profile function
$\unicode[STIX]{x1D719}:[-1,1]\times G\rightarrow \mathbb{C}$
satisfying certain continuity and integrability assumptions, we show that the positive definiteness of the kernel
$((x,u),(y,v))\in (\mathbb{H}\times G)^{2}\mapsto \unicode[STIX]{x1D719}(\cos \unicode[STIX]{x1D6FF}(x,y),uv^{-1})$
is equivalent to the positive definiteness of the Fourier transformed kernels
$(x,y)\in \mathbb{H}^{2}\mapsto \widehat{\unicode[STIX]{x1D719}}_{\cos \unicode[STIX]{x1D6FF}(x,y)}(\unicode[STIX]{x1D6FE})$
,
$\unicode[STIX]{x1D6FE}\in \widehat{G}$
, where
$\unicode[STIX]{x1D719}_{t}(u)=\unicode[STIX]{x1D719}(t,u)$
,
$u\in G$
. We also provide some results on the strict positive definiteness of the kernel.