Schoenberg's theorem for positive definite functions on products: a unifying framework (Q2003595)

The authors present a unified approach for positive definiteness and strict positive definiteness on a product of the form \(X\times S^d\), (\(d=1,2, \dots\)), in which \(X\) is a nonempty set and \(S^d\) is the usual \(d\)-dimensional unit sphere in the Euclidean space, fixing no algebraic or topological structures on \(X\), but keeping the continuity and isotropy of \(S^d\). The main result may be regarded as an extension of the classical result of \textit{I. J. Schoenberg} [Duke Math. J. 9, 96--108 (1942; Zbl 0063.06808)]. They characterize positive definiteness and strict positive definiteness, and indicate how to recover some known cases from their results. They also examine their results in the case when \(X\) is a locally compact group, the group \((\mathbb{R}^m,+)\), and the \(m\)-dimensional torus.