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.

在本文中，我们考虑了在紧致两点齐次空间上的局部正则阿贝尔群上刻画正定函数的问题。对于具有双组 $ \ widehat {G} $ 的局部紧致阿贝尔群 $ G $ ， 具有规范化测地距离 $ \ unicode [STIX] {x1D6FF} $ 和一个a的紧致两点齐次空间 $ \ mathbb {H} $ 轮廓函数 $ \ unicode [STIX] {x1D719}：[-1,1] \ timesG \ rightarrow \ mathbb {C} $ 满足某些连续性和可积性假设，我们证明了内核 $（（x， u），（y，v））\ in（\ mathbb {H} \ times G）^ {2} \ mapsto \ unicode [STIX] {x1D719}（\ cos \ unicode [STIX] {x1D6FF}（x，y ），uv ^ {-1}）$ 等效 于\ mathbb {H} ^ {2} \ mapsto \ widehat {\ unicode [STIX] {x1D719}} _ {\ cos \ unicode [STIX]中的傅立叶变换内核$（x，y）\ ] {x1D6FF}（x，y）}（\ unicode [STIX] {x1D6FE}）$ ， $ \ unicode [STIX] {x1D6FE} \ in \ widehat {G} $中 ，其中 $ \ unicode [STIX] {x1D719} _ {t}（u）= \ unicode [STIX] {x1D719}（t，u）$ ， $ u \ in G $ 。我们还提供了关于内核的严格正定性的一些结果。