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 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.

翻译：本文定义了域上的正定函数与严格正定函数，并在一系列正则域上研究了这些函数。该系列包括单位球、锥面、双曲面、实心双曲面体以及单纯形。每个正则域通过保距映射嵌入到单位球面的一个卦限或若干卦限的并集中，由此推导出这些正则域上正定函数与严格正定函数的刻画。