We give a characterization of positive definite integrable functions on a product of two Gelfand pairs as an integral of positive definite functions on one of the Gelfand pairs with respect to the Plancherel measure on the dual of the other Gelfand pair. In the very special case where the Gelfand pairs are Euclidean groups and the compact subgroups are reduced to the identity, the characterization is a much cited result in spatio-temporal statistics due to Cressie, Huang and Gneiting. When one of the Gelfand pairs is compact the characterization leads to results about expansions in spherical functions with positive definite expansion functions, thereby recovering recent results of the author in collaboration with Peron and Porcu. In the special case when the compact Gelfand pair consists of orthogonal groups, the characterization is important in geostatistics and covers a recent result of Porcu and White.

我们将两个Gelfand对的乘积上的正定可积函数的特征描述为，一个Gelfand对上的正定函数相对于另一个Gelfand对的对偶上的Plancherel测度的积分。在非常特殊的情况下，Gelfand对是欧几里得组，而紧凑的亚组被还原为恒等式，由于Cressie，Huang和Gneiting的影响，表征是时空统计中被广泛引用的结果。当Gelfand对中的一个紧密时，表征将得出具有正定膨胀函数的球形函数膨胀的结果，从而与Peron和Porcu合作恢复了作者的最新结果。在特殊情况下，当紧凑的Gelfand对由正交组组成时，