On the extremal rays of the cone of positive, positive definite functions

Details On the extremal rays of the cone of positive, positive definite functions On the extremal rays of the cone of positive, positive definite functions

2008-01-07

arxiv.org/abs/0801.0941v1

Journal of fourier analysis and applications 15 (2009) 561-582

Mathematics

Classical Analysis and ODEs

Scientific paper

10.1007/s00041-008-9057-6

The aim of this paper is to investigate the cone of non-negative, radial, positive-definite functions in the set of continuous functions on $\R^d$. Elements of this cone admit a Choquet integral representation in terms of the extremals. The main feature of this article is to characterize some large classes of such extremals. In particular, we show that there many other extremals than the gaussians, thus disproving a conjecture of G. Choquet and that no reasonable conjecture can be made on the full set of extremals. The last feature of this article is to show that many characterizations of positive definite functions available in the literature are actually particular cases of the Choquet integral representations we obtain.

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