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Results in Mathematics

, Volume 26, Issue 3–4, pp 229–237 | Cite as

Functional Equations and Distribution Functions

  • Jonathan M. Borwein
  • Roland Girgensohn
Article

Abstract

We consider the functional equation
$$f(t)={1\over b}{\mathop \sum^{b-1}\limits_{\nu=0}}f\Bigg({t-\beta_{\nu}\over a}\Bigg)\ \ \ {\rm for\ all}\ t\in {\rm R},$$
where 0 < a < 1, b in ℕ {1} and −1 = β 0 ≤ β1 ≤ … ≤ βb− 1 =1 are given parameters, ƒ: ℝ → ℝ is the unknown. We show that there is a unique bounded function ƒ which solves (F) and satisfies ƒ(t) = 0 for t < ∼-1/(1 − a), ƒ(t) = 1 for t > 1/(1 − a). This solution can be interpreted as the distribution function of a certain random series. It is known to be either singular or absolutely continuous, but the problem for which parameters it is absolutely continuous is largely open. We collect some previously established partial answers and generalize them. We also point out an interesting connection to the so-called Schilling equation.

AMS (1991) subject classification

39B22 62E15 39B62 60E05 

Key words

Distribution functions Bernoulli distributions singular functions Schilling equation Vieta’s product 

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References

  1. [1]
    K. Baron, A. Simon, P. Volkmann, Solutions d’une équation fonctionnelle dans l’espace des distributions tempérées, Preprint. Google Scholar
  2. [2]
    M. J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse, J. P. Schreiber, Pisot and Salem numbers, Birkhäuser, Basel 1992.MATHCrossRefGoogle Scholar
  3. [3]
    P. Erdős, On a family of symmetric Bernoulli convolutions, Trans. Amer. Math. Soc. 61 (1939), 974–976.Google Scholar
  4. [4]
    P. Erdős, On the smoothness properties of a family of Bernoulli convolutions, Amer. J. Math. 62 (1940), 180–186.MathSciNetCrossRefGoogle Scholar
  5. [5]
    W. Förg-Rob, On a problem of R. Schilling, Math. Pannonica, to appear.Google Scholar
  6. [6]
    A. Garsia, Arithmetic properties of Bernoulli convolutions, Trans. Amer. Math. Soc. 102 (1962), 409–432.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    P. R. Halmos, Measure Theory, Van Nostrand Reinhold, New York 1950.MATHGoogle Scholar
  8. [8]
    H. Helson, Harmonic Analysis, Addison-Wesley, Reading 1983.MATHGoogle Scholar
  9. [9]
    B. Jessen, A. Wintner, Distribution functions and the Riemann zeta function, Trans. Amer. Math. Soc. 38 (1935), 48–88.MathSciNetCrossRefGoogle Scholar
  10. [10]
    M. Kac, Statistical Independence in Probability, Analysis and Number Theory, Wiley 1959Google Scholar
  11. [11]
    R. Kershner, A. Wintner, On symmetric Bernoulli convolutions, Amer. J. Math. 57 (1935), 541–548.MathSciNetCrossRefGoogle Scholar
  12. [12]
    J. Morawiec, On bounded solutions of a problem of R. Schilling, Preprint. Google Scholar
  13. [13]
    S. Paganoni Marzegalli, One-parameter system of functional equations, Aequationes Math. 47 (1994), 50–59.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    R. Schilling, Spatially chaotic strucutres, in: H. Thomas (editor), Nonlinear Dynamics in Solids, Springer, Berlin 1992, 213–241.CrossRefGoogle Scholar
  15. [15]
    A. Wintner, On convergent Poisson convolutions, Amer. J. Math. 57 (1935), 827–838.MathSciNetCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag, Basel 1994

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsSimon Fraser UniversityBurnabyCanada

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