## Abstract

In his seminal paper, Schoenberg (Duke Math J 9:96–108, 1942) characterized the class P(S^{d}) of continuous functions f: [- 1 , 1] → R such that f(cos θ(ξ, η)) is positive definite on the product space S^{d}× S^{d}, with S^{d} being the unit sphere of R^{d} ^{+} ^{1} and θ(ξ, η) being the great circle distance between ξ, η∈ S^{d}. In the present paper, we consider the product space S^{d}× G, for G a locally compact group, and define the class P(S^{d}, G) of continuous functions f: [- 1 , 1] × G→ C such that f(cos θ(ξ, η) , u^{- 1}v) is positive definite on S^{d}× S^{d}× G× G. This offers a natural extension of Schoenberg’s theorem. Schoenberg’s second theorem corresponding to the Hilbert sphere S^{∞} is also extended to this context. The case G= 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.

Original language | British English |
---|---|

Pages (from-to) | 217-241 |

Number of pages | 25 |

Journal | Constructive Approximation |

Volume | 45 |

Issue number | 2 |

DOIs | |

State | Published - 1 Apr 2017 |

## Keywords

- Positive definite
- Space-time covariances
- Spherical harmonics