Orthogonal expansions related to compact Gelfand pairs

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∈ZB(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∈ZB(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).