Results in Mathematics

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

Functional Equations and Distribution Functions

  • Jonathan M. BorweinEmail author
  • Roland Girgensohn


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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Copyright information

© Birkhäuser Verlag, Basel 1994

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsSimon Fraser UniversityBurnabyCanada

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