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Lookup NU author(s): Professor Emilio Porcu
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND).
© 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×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 ∑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)<∞. 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 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).
Author(s): Berg C, Peron AP, Porcu E
Publication type: Article
Publication status: Published
Journal: Expositiones Mathematicae
Year: 2017
Volume: 36
Issue: 3-4
Pages: 259-277
Print publication date: 01/12/2018
Online publication date: 09/11/2017
Acceptance date: 02/04/2016
Date deposited: 05/02/2018
ISSN (print): 0723-0869
ISSN (electronic): 1878-0792
Publisher: Elsevier GmbH - Urban und Fischer
URL: https://doi.org/10.1016/j.exmath.2017.10.005
DOI: 10.1016/j.exmath.2017.10.005
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