On the Extremal Rays of the Cone of Positive, Positive Definite Functions

Mathematics - Functional Analysis QA Mathematics / matematika Mathematics - Classical Analysis and ODEs Probability (math.PR) 42A82 Classical Analysis and ODEs (math.CA) FOS: Mathematics 0101 mathematics 01 natural sciences Mathematics - Probability Functional Analysis (math.FA)
DOI: 10.1007/s00041-008-9057-6 Publication Date: 2009-01-14T16:47:01Z
ABSTRACT
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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