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In his seminal paper, Schoenberg (Duke Math J 9:96-108, 1942) characterized the class $$\mathcal P(\mathbb {S}^d)$$ of continuous functions $$f:[-1,1] \rightarrow \mathbb {R}$$ such that $$f(\cos \theta (\xi ,\eta ))$$ is positive definite on the product space $$\mathbb {S}^d \times \mathbb {S}^d$$ , with $$\mathbb {S}^d$$ being the unit sphere of $$\mathbb {R}^{d+1}$$ and $$\theta (\xi ,\eta )$$ being the great circle distance between $$\xi ,\eta \in \mathbb {S}^d$$ . In the present paper, we consider the product space $$\mathbb {S}^d \times G$$ , for G a locally compact group, and define the class $$\mathcal P(\mathbb {S}^d, G)$$ of continuous functions $$f:[-1,1]\times G \rightarrow \mathbb {C}$$ such that $$f(\cos \theta (\xi ,\eta ), u^{-1}v)$$ is positive definite on $$\mathbb {S}^d \times \mathbb {S}^d \times G \times G$$ . This offers a natural extension of Schoenberg's theorem. Schoenberg's second theorem corresponding to the Hilbert sphere $$\mathbb {S}^\infty $$ is also extended to this context. The case $$G=\mathbb {R}$$ 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. [ABSTRACT FROM AUTHOR]