Skip to main content
SpringerLink
Log in

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

  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Akemann, C.A., Walter, M.E.: Non-Abelian Pontriagin duality. Duke Math. J. 39, 17–38 (1972)

    Article  MathSciNet  Google Scholar 

  2. Ball, K.: Completely monotonic rational functions and Hall’s marriage theorem. J. Comb. Theory, Ser. B 61, 118–124 (1994)

    Article  MATH  Google Scholar 

  3. Baxter, B.J.C.: Positive definite functions on Hilbert space. East J. Approx. 10, 269–274 (2004)

    MATH  MathSciNet  Google Scholar 

  4. Becker, R.: Convex Cones in Analysis. Travaux en cours, vol. 67. Hermann, Paris (2006)

    Google Scholar 

  5. Berg, C., Forst, G.: Potential Theory on Locally Compact Abelian Groups. Ergebnisse der Math., vol. 87. Springer, Berlin (1975)

    MATH  Google Scholar 

  6. Berg, C., Christensen, J.P.R., Ressel, P.: Harmonic Analysis on Semigroups. Theory of Positive Definite and Related Functions. Graduate Texts in Mathematics, vol. 100. Springer, Berlin (1984)

    MATH  Google Scholar 

  7. Berg, C., Mateu, J., Porcu, E.: The Dagum family of isotropic correlation functions. Available at arXiv:0705.0456v1 [math.ST]

  8. Bochner, S.: Monotone Funktionen, Stieltjessche Integrale und harmonische Analyse. Math. Ann. 108, 378–410 (1933)

    Article  MathSciNet  Google Scholar 

  9. Bondesson, L.: Generalized Gamma Convolutions and Related Classes of Distributions and Densities. Lecture Notes in Statistics, vol. 76. Springer, New York (1992)

    MATH  Google Scholar 

  10. Borisov, A.: Positive positive definite functions on Abelian groups. Preprint (1999). Available on arXiv:math/9906126v1

  11. Bonami, A., Demange, B., Jaming, Ph.: Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms. Rev. Mat. Iberoam. 19, 23–55 (2003)

    MATH  MathSciNet  Google Scholar 

  12. Bucy, R.S., Maletese, G.: Extreme positive definite functions and Choquet’s representation theorem. J. Math. Anal. Appl. 12, 371–377 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  13. Buhmann, M.D.: Radial Basis Functions: Theory and Implementations. Cambridge Monographs on Applied and Computational Mathematics, vol. 12. Cambridge University Press, Cambridge (2003)

    MATH  Google Scholar 

  14. Choquet, G.: Deux exemples classiques de représentation intégrale. Enseign. Math. 15, 63–75 (1969)

    MATH  MathSciNet  Google Scholar 

  15. Choquet, G.: Lectures on Analysis. Benjamin, Elmsford (1969). Marsen, J., Lance, T., Gelbart, S. (eds.)

    Google Scholar 

  16. Day, W.A.: On monotonicity of the relaxation functions of viscoelastic material. Proc. Camb. Philos. Soc. 67, 503–508 (1970)

    Article  MATH  Google Scholar 

  17. Demange, B.: Principes d’incertitude associés à des formes quadratiques non dégénérées. Thèse de l’université d’Orléans (2004)

  18. Demange, B.: Uncertainty principles related to quadratic forms. Mem. Soc. Math. Fr. (2009, to appear)

  19. Feller, W.: An Introduction to Probability Theory and Applications. Willey, New York (1971)

    MATH  Google Scholar 

  20. Folland, G.B.: A Course in Abstract Harmonic Analysis. Studies in Advanced Mathematics. CRC Press, Boca Raton (1995)

    Google Scholar 

  21. Folland, G.B., Sitaram, A.: The uncertainty principle: a mathematical survey. J. Fourier Anal. Appl. 3, 207–238 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  22. Frenzen, C.L.: Error bounds for asymptotic expansions of the ratio of two gamma functions. SIAM J. Math. Anal. 18, 890–896 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  23. Giraud, B.G., Peschanski, R.: On positive functions with positive Fourier transforms. Acta Phys. Pol. B 37, 331–346 (2006)

    MathSciNet  Google Scholar 

  24. Gneiting, T.: Kuttner’s problem and a Pólya type criterion for characteristic functions. Proc. Am. Math. Soc. 128, 1721–1728 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  25. Gneiting, T.: Criteria of Pólya type for radial positive definite functions. Proc. Am. Math. Soc. 129, 2309–2318 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  26. Gneiting, T., Schlather, M.: Stochastic models that separate fractal dimension and the Hurst effect. SIAM Rev. 46, 269–282 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  27. Godefroy, G.: Personal communication

  28. Grafakos, L.: Classical and Modern Fourier Analysis. Prentice Hall, New York (2004)

    MATH  Google Scholar 

  29. Hainzl, C., Seiringer, R.: General decomposition of radial functions on ℝn and applications to N-body quantum systems. Lett. Math. Phys. 61, 75–84 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  30. Hardy, G.H.: A theorem concerning Fourier transforms. J. Lond. Math. Soc. 8, 227–231 (1933)

    Article  MATH  Google Scholar 

  31. Havin, V., Jöricke, B.: The Uncertainty Principle in Harmonic Analysis. Springer, Berlin (1994)

    MATH  Google Scholar 

  32. Hurwitz, A.: Über die Nullstellen der Bessel’schen Funktion. Math. Ann. 33, 246–266 (1889). See Mathematische Werke 1, 266–286 (1932)

    Article  Google Scholar 

  33. Kimberling, C.H.: A probabilistic interpretation of complete monotonicity. Aequ. Math. 10, 152–164 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  34. Kaniuth, E., Lau, A.T.: A separation property of positive definite functions on locally compact groups and applications to Fourier algebras. J. Funct. Anal. 175, 89–110 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  35. Kaniuth, E., Lau, A.T.: Extension and separation properties of the positive definite functions on locally compact groups. Trans. Am. Math. Soc. 359, 447–463 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  36. Koldobsky, A.: Fourier Analysis in Convex Geometry. Mathematical Surveys and Monographs, vol. 116. Am. Math. Soc., Providence (2005)

    MATH  Google Scholar 

  37. Lévy, P.: Fonctions caractéristiques positives (Positive characteristic functions). C.R. Acad. Sci. Paris, Sér. A–B 265, 249–252 (1967)

    MATH  Google Scholar 

  38. Mateu, J., Porcu, E., Nicolis, O.: A note on decoupling of local and global behaviour for the Dagum random field. Probab. Eng. Mech. 28, 320–329 (2007)

    Article  Google Scholar 

  39. Phelps, R.: Lectures on Choquet’s Theorem. Van Nostrand, Princeton (1966)

    MATH  Google Scholar 

  40. Pólya, G.: Remarks on characteristic functions. In: Neyman, J. (ed.) Proceedings of the Berkeley Symposium on Mathematical Statistics and Probability, pp. 115–123. University of California Press, Berkeley (1949)

    Google Scholar 

  41. Porcu, E., Mateu, J., Zini, A., Pini, R.: Modelling spatiotemporal data: a new variogram and covariance structure proposal. Stat. Probab. Lett. 77, 83–89 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  42. Sasvári, Z.: On a classical theorem in the theory of Fourier integrals. Proc. Am. Math. Soc. 126, 711–713 (1998)

    Article  MATH  Google Scholar 

  43. Schoenberg, I.J.: Metric spaces and completely monotone functions. Ann. Math. 39, 811–841 (1938)

    Article  MathSciNet  Google Scholar 

  44. Schwartz, L.: Théorie des distributions. Vol. II. Actualités Sci. Indust., vol. 1122. Hermann, Paris (1951)

    Google Scholar 

  45. Steerneman, A.G.M., van Perlo-ten Kleij, F.: Spherical distributions: Schoenberg (1938) revisited. Exp. Math. 23, 281–287 (2005)

    MATH  Google Scholar 

  46. Steward, J.: Positive definite functions and generalizations, an historical survey. Rocky Mt. J. Math. 6, 409–434 (1976)

    Article  Google Scholar 

  47. Thomas, E.G.F.: Bochner and Bernstein theorems via the nuclear integral representation theorem. Special issue dedicated to John Horváth. J. Math. Anal. Appl. 297, 612–624 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  48. Watson, G.N.: A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press, Cambridge (1944)

    MATH  Google Scholar 

  49. Wendland, H.: Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. J. Convex Anal. 4, 389–396 (1995)

    MATH  MathSciNet  Google Scholar 

  50. Widder, D.V.: The Laplace Transform. Princeton Univ. Press, Princeton (1941)

    MATH  Google Scholar 

  51. Wimp, J.: Sequence Transformations and Their Applications. Academic Press, San Deigo (1981)

    MATH  Google Scholar 

  52. Wu, Z.: Compactly supported positive definite radial functions. Adv. Comput. Math. 4, 283–292 (1995)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire MAPMO, CNRS, UMR 6628, Université d’Orléans, Fédération Denis Poisson, FR 2964, Bâtiment de Mathématiques BP 6759, 45067, Orleans cedex 2, France

    Philippe Jaming

  2. Rényi Institute of Mathematics, 1053, Reáltanoda u. 13-15, Budapest, Hungary

    Máté Matolcsi & Szilárd G. Révész

  3. BME Department of Analysis, Egry J. u. 1, Budapest, 1111, Hungary

    Máté Matolcsi

Authors
  1. Philippe Jaming
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Máté Matolcsi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Szilárd G. Révész
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Philippe Jaming.

Additional information

Communicated by Gerald B. Folland.

The authors wish to thank E.G.F. Thomas for valuable conversations and G. Godefroy for mentioning the problem to them.

M. Matolcsi was supported by Hungarian National Foundation for Scientific Research (OTKA), Grants No. PF-64061, T-04930, and by a Visiting Professorship of the University of Orleans, October 2007.

The third author was supported by Hungarian National Science Foundation, grant no.’s OTKA # T 049301, K 061908 and K 072731.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jaming, P., Matolcsi, M. & Révész, S.G. On the Extremal Rays of the Cone of Positive, Positive Definite Functions. J Fourier Anal Appl 15, 561–582 (2009). https://doi.org/10.1007/s00041-008-9057-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00041-008-9057-6

Keywords

Mathematics Subject Classification (2000)

Search

Navigation