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

A unified view of space-time covariance functions through Gelfand pairs