## Abstract

For a locally compact group G, let P(G) denote the set of continuous positive definite functions [Formula presented]. Given a compact Gelfand pair [Formula presented] and a locally compact group L, we characterize the class [Formula presented] of functions f∈P(G×L) which are bi-invariant in the G-variable with respect to K. The functions of this class are the functions having a uniformly convergent expansion ∑_{φ∈Z}B(φ)(u)φ(x) for [Formula presented], where the sum is over the space Z of positive definite spherical functions [Formula presented] for the Gelfand pair, and (B(φ))_{φ∈Z} is a family of continuous positive definite functions on L such that ∑_{φ∈Z}B(φ)(e_{L})<∞. Here e_{L} is the neutral element of the group L. For a compact Abelian group G considered as a Gelfand pair [Formula presented] with trivial K = {e_{G}}, we obtain a characterization of P(G×L) in terms of Fourier expansions on the dual group [Formula presented]. The result is described in detail for the case of the Gelfand pairs [Formula presented] and [Formula presented] as well as for the product of these Gelfand pairs. The result generalizes recent theorems of Berg–Porcu (2016) and Guella–Menegatto (2016).

Original language | British English |
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Pages (from-to) | 259-277 |

Number of pages | 19 |

Journal | Expositiones Mathematicae |

Volume | 36 |

Issue number | 3-4 |

DOIs | |

State | Published - Dec 2018 |

## Keywords

- Gelfand pairs
- Positive definite functions
- Spherical functions
- Spherical harmonics for real an complex spheres