Orthogonal expansions related to compact Gelfand pairs

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 PK♯(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∈ZB(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∈ZB(φ)(eL)<∞. Here eL is the neutral element of the group L. For a compact Abelian group G considered as a Gelfand pair (G,K) with trivial K={eG}, we obtain a characterization of P(G×L) in terms of Fourier expansions on the dual group Ĝ.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).