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From Schoenberg Coefficients to Schoenberg Functions

Lookup NU author(s): Professor Emilio Porcu

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Abstract

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(cos⁡θ(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(cos⁡θ(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.

Publication metadata

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

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