From Schoenberg Coefficients to Schoenberg Functions

In his seminal paper, Schoenberg (Duke Math J 9:96-108, 1942) characterized the class of continuous functions such that is positive definite on the product space , with being the unit sphere of and being the great circle distance between . In the present paper, we consider the product space , for G a locally compact group, and define the class of continuous functions such that is positive definite on . This offers a natural extension of Schoenberg's theorem. Schoenberg's second theorem corresponding to the Hilbert sphere is also extended to this context. The case is of special importance for probability theory and stochastic processes, because it characterizes completely the class of space-time covariance functions where the space is the sphere, being an approximation of planet Earth.