From Riesz Products to Random Sets

  • Jean-Pierre Kahane
Conference paper


This talk consists of three interrelated parts: I. Riesz products; II. Random operators coming from multiplication of independent factors; III. Random coverings. The linkage is the role of singular measures and their dimensional analysis.


Random Measure Trigonometric Series Multifractal Analysis Finite Energy Random Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    P. BILLARD Series de Fourier aleatoirement bornees, continues, uniformement con- vergentes. Ann. Scient. Ec. Norm. Sup. 82 (1965), 131–179.MathSciNetMATHGoogle Scholar
  2. [2]
    G. BROWN & W. MORAN On orthogonality of Riesz products. Proc. Cambridge Philos. Soc. 76 (1974), 173–181.MathSciNetMATHGoogle Scholar
  3. [3]
    A. DVORETZKY On covering a circle by randomly placed arcs. Proc. Nat. Acad. Sc. USA 42 (1956), 199–203.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Y. EL HELOU Recouvrement du tore Tq par des ouverts aleatoires et dimension de Vensemble non recouvert. C. R. Acad. Sc. Paris 287 A (1978), 8152–818.Google Scholar
  5. [5]
    FAN Ai-Hua Une condition sufEsante d’existence du moment d’ordre m (entier) du chaos multiplicatif. Ann. Sc. Math. Quebec 10 (1986), 119–120.MATHGoogle Scholar
  6. [6]
    FAN Ai-Hua These, Orsay 1989.Google Scholar
  7. [7]
    FAN Ai-Hua Chaos additif et chaos multiplicatif de Levy. C. R. Acad. Sc. Paris 308 (1989) , 151–154.MATHGoogle Scholar
  8. [8]
    FAN Ai-Hua Sur la convergence de series trigonometriques lacunaires presque partout par rapport a des produits de Riesz. C. R. Acad. Sc. Paris 309 (1989), 295–298.MATHGoogle Scholar
  9. [9]
    J. HOFFMAN-JORGENSEN Coverings of metric spaces by randomly placed balls. Math. Scand. 32 (1973), 169–186.MathSciNetGoogle Scholar
  10. [10]
    J.-P. KAHANE Some random series of functions. Cambridge Univ. Press 1985.Google Scholar
  11. [11]
    J.-P. KAHANE Surle chaos multiplicatif. Ann. Sc. Math. Quebec 9 (1985), 105–150 and 10 (1986), 117–118.MathSciNetMATHGoogle Scholar
  12. [12]
    J.-P. KAHANE Positive martingales and random measures. Chin. Ann. Math. 8 B1 (1987), 1–12.Google Scholar
  13. [13]
    J.-P. KAHANE Intervalles aleatoires et decomposition des mesures. C. R. Acad. Sc. Paris 304 (1987), 551–554.MATHGoogle Scholar
  14. [14]
    J.-P. KAHANE Produits de poids aleatoires independants et applications. Cours a PUniv. Montreal, prepubl. Orsay 89–33.Google Scholar
  15. [15]
    J.-P. KAHANE Recouvrement par des simplexes homothetiques aleatoires. C. R. Acad. Sc. Paris 310 (1990), 419–423.MATHGoogle Scholar
  16. [16]
    J.-P. KAHANE Recouvrements aleatoires et theorie du potentiel. Coll. Math. 50 (1990) , 1–25.Google Scholar
  17. [17]
    J.-P. KAHANE & J. PEYRIERE Sur certaines martingales de Benoit Mandelbrot Advances in Math. 22 (1976), 131–145.Google Scholar
  18. [18]
    S. J. KILMER k. S. SAEKI On Riesz product measures; mutual absolute continuity and singularity. Ann. Inst. Fourier, Grenoble, 38 (1988), 63–93.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    B. B. MANDELBROT Possible refinement of the log-normal hypothesis concerning the distribution of energy dissipation in intermittent turbulence. Lecture Notes in Physics, Springer Verlag 1972, 333–351.Google Scholar
  20. B. B. MANDELBROT Multiplications aleatoires iterees et distributions invariantes par moyenne ponderee aleatoire. C.R.Acad. Sc. Paris 278 (1974), 289–292 et 355–358.MathSciNetMATHGoogle Scholar
  21. [21]
    F. PARREAU Ergodicite et purete des produits de Riesz. Ann. Inst. Fourier 40 (1990), 391–405.MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    J. PEYRIERE Surles produits de Riesz. C. R. Acad. Sc. Paris A-B 276 (1973), 1417–1419.MathSciNetMATHGoogle Scholar
  23. [23]
    J. PEYRIERE Etude de quelques proprietes des produits de Riesz. Ann. Inst. Fourier (Grenoble), 25 (1975), 127–169.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    J. PEYRIERE Almost everywhere convergence of lac unary trigonometric series with respect to Riesz products. Australian J. Math, (to appear).Google Scholar
  25. [25]
    F. RIESZ Uber die FourierkoefEzienten einer stetigen Funktion von beschrankter Schwankung. M. Zeitschrift 2 (1918), 312–315.MathSciNetCrossRefGoogle Scholar
  26. [26]
    R. SALEM On some singular monotonic functions which are strictly increasing. Trans. Amer. Math. Soc. 53 (1943), 427–439.MathSciNetMATHCrossRefGoogle Scholar
  27. [27]
    L. SHEPP Covering the circle with random arcs. Israel J. Math. 11 (1972), 328–345.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Tokyo 1991

Authors and Affiliations

  • Jean-Pierre Kahane
    • 1
  1. 1.MathématiquesUniversité de Paris-SudOrsay CedexFrance

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