Browse by author
Lookup NU author(s): Professor Emilio Porcu
Full text for this publication is not currently held within this repository. Alternative links are provided below where available.
In his seminal paper, Schoenberg (Duke Math J 9:96–108, 1942) characterized the class P(S^d) of continuous functions f:[−1,1]→R such that f(cosθ(ξ,η)) is positive definite on the product space Sd×Sd, with Sd being the unit sphere of Rd+1 and θ(ξ,η) being the great circle distance between ξ,η∈Sd . In the present paper, we consider the product space Sd×G, for G a locally compact group, and define the class P(Sd,G) of continuous functions f:[−1,1]×G→C such that f(cosθ(ξ,η),u−1v) is positive definite on Sd×Sd×G×G. This offers a natural extension of Schoenberg’s theorem. Schoenberg’s second theorem corresponding to the Hilbert sphere S∞ is also extended to this context. The case G=R is of special importance for probability theory and stochastic processes, because it characterizes completely the class of space-time covariance functions where the space is the sphere, being an approximation of planet Earth.
Author(s): Berg C, Porcu E
Publication type: Article
Publication status: Published
Journal: Constructive Approximation
Year: 2017
Volume: 45
Issue: 2
Pages: 217–241
Print publication date: 01/04/2017
Online publication date: 21/01/2016
Acceptance date: 23/11/2015
ISSN (print): 0176-4276
ISSN (electronic): 1432-0940
Publisher: Springer
URL: http://doi.org/10.1007/s00365-016-9323-9
DOI: 10.1007/s00365-016-9323-9
Altmetrics provided by Altmetric