| Full text | |
| Author(s): |
Berg, Christian
;
Peron, Ana P.
;
Porcu, Emilio
Total Authors: 3
|
| Document type: | Journal article |
| Source: | EXPOSITIONES MATHEMATICAE; v. 36, n. 3-4, p. 19-pg., 2018-01-01. |
| 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 P-K(#)(G L) of functions f is an element of P(G x 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 Sigma(phi is an element of Z) B(phi)(u)phi(x) for x is an element of G, u is an element of L, where the sum is over the space Z of positive definite spherical functions phi : G -> C for the Gelfand pair, and (B(phi))(phi is an element of Z) is a family of continuous positive definite functions on L such that E Sigma(phi is an element of Z) B(phi)(e(L)) < infinity. Here e(L) is the neutral element of the group L. For a compact Abelian group G considered as a Gelfand pair (G, K) with trivial K = {e(G)}, we obtain a characterization of P(G x L) in terms of Fourier expansions on the dual group <(G)over cap>. 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). (C) 2017 Elsevier GmbH. All rights reserved. (AU) | |
| FAPESP's process: | 16/03015-7 - Positive definite functions |
| Grantee: | Ana Paula Peron |
| Support Opportunities: | Scholarships abroad - Research |
| FAPESP's process: | 14/25796-5 - (Strict) positive definite functions and differentiability |
| Grantee: | Ana Paula Peron |
| Support Opportunities: | Regular Research Grants |