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
AUTHORS (3)
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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