LAUSR

Abstract For a locally compact group G , let P ( G ) denote the set of continuous positive definite functions f : G → C . Given a compact Gelfand pair ( G , K ) and a locally compact group L , we characterize the class P K ♯ ( G , L ) 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 x ∈ G , u ∈ L , where the sum is over the space Z of positive definite spherical functions φ : G → C 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 ( G , K ) with trivial K = { e G } , we obtain a characterization of P ( G × L ) in terms of Fourier expansions on the dual group G . The result is described in detail for the case of the Gelfand pairs ( O ( d + 1 ) , O ( d ) ) and ( U ( q ) , U ( q − 1 ) ) as well as for the product of these Gelfand pairs. The result generalizes recent theorems of Berg–Porcu (2016) and Guella–Menegatto (2016).