Schoenberg’s Theorem for Positive Definite Functions on Products: A Unifying Framework

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The main contribution in the present paper is a characterization for positive definiteness and strict positive definiteness of a kernel on the product $$X \times S^d$$ , in which X is a nonempty set and $$S^d$$ is the usual d-dimensional unit sphere in Euclidean space, through Fourier-like expansions. The setting presupposes continuity and isotropy on the $$S^d$$ side and no algebraic structure or topology on X. The result may be interpreted as another extension of a classical result of I. J. Schoenberg on positive definite functions on spheres. We take a closer look at our results in the case in which X is a locally compact group, paying special attention to usual Euclidean spaces and high dimensional tori.