The aim of this paper is to investigate the cone of non-negative, radial, positive-definite functions in the set of continuous functions on ℝd. 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 are 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.
- Choquet integral representation
- Extremal ray generators
- Positive definite functions
ASJC Scopus subject areas
- Applied Mathematics