We define positive and strictly positive definite functions on a domain and study these functions on a list of regular domains. The list includes the unit ball, conic surface, hyperbolic surface, solid hyperboloid and the simplex. Each of these domains is embedded in a quadrant or a union of quadrants of the unit sphere by a distance-preserving map, from which characterizations of positive definite and strictly positive definite functions are derived for these regular domains.

中文翻译：

---

正则域上的正定函数


我们在一个域上定义正和严格正定函数，并在常规域列表中研究这些函数。该列表包括单位球、圆锥曲面、双曲曲面、实心双曲面和单纯形。这些域中的每一个都通过距离守恒映射嵌入到单位球体的象限或象限联合中，从中可以推导出这些规则域的正定函数和严格正定函数的特征。