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Journal of Fourier Analysis and Applications

, Volume 15, Issue 4, pp 561–582 | Cite as

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

  • Philippe JamingEmail author
  • Máté Matolcsi
  • Szilárd G. Révész
Article

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 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.

Keywords

Choquet integral representation Extremal ray generators Positive definite functions 

Mathematics Subject Classification (2000)

42A82 

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Copyright information

© Birkhäuser Boston 2009

Authors and Affiliations

  • Philippe Jaming
    • 1
    Email author
  • Máté Matolcsi
    • 2
    • 3
  • Szilárd G. Révész
    • 2
  1. 1.Laboratoire MAPMO, CNRS, UMR 6628Université d’OrléansOrleans cedex 2France
  2. 2.Rényi Institute of MathematicsBudapestHungary
  3. 3.BME Department of Analysis, Egry J. u. 1BudapestHungary

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