Orthogonal expansions related to compact Gelfand pairs

Christian Berg, Ana P. Peron, Emilio Porcu

Research output: Contribution to journalArticlepeer-review

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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)(eL)<∞. Here eL is the neutral element of the group L. For a compact Abelian group G considered as a Gelfand pair [Formula presented] with trivial K = {eG}, 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 languageBritish English
Pages (from-to)259-277
Number of pages19
JournalExpositiones Mathematicae
Issue number3-4
StatePublished - Dec 2018


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


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