Berg C

Peron AP

Porcu E

\ua9 2017 Elsevier GmbH. 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 fεP(G
7L) 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 ∑ϕεZB(ϕ)(u)ϕ(x) for xεG,uεL, where the sum is over the space Z of positive definite spherical functions ϕ:G→C for the Gelfand pair, and (B(ϕ))ϕεZ is a family of continuous positive definite functions on L such that ∑ϕεZB(ϕ)(eL)&lt;∞. 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
7L) 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)

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Orthogonal expansions related to compact Gelfand pairs

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Orthogonal expansions related to compact Gelfand pairs

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