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

Philippe Jaming, Máté Matolcsi, Szilárd G. Révész

Research output: Contribution to journalArticle

5 Citations (Scopus)

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 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.

Original languageEnglish
Pages (from-to)561-582
Number of pages22
JournalJournal of Fourier Analysis and Applications
Volume15
Issue number4
DOIs
Publication statusPublished - Oct 1 2009

Keywords

  • Choquet integral representation
  • Extremal ray generators
  • Positive definite functions

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)
  • Applied Mathematics

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