Functional Equations and Distribution Functions

Borwein, Jonathan M. and Girgensohn, Roland (1994) Functional Equations and Distribution Functions. Results in Mathematics, 26 (3-4). pp. 229-237.

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We consider the functional equation $\ds f(t) = {1 \over b} \sum_{\nu=0}^{b-1} f\left({t-\beta_\nu \over a}\right) \quad \mbox{for all } t \in \R,$ where $0<a<1$, $b \in \N \setminus \{1\}$ and $-1 = \beta_0 \le \beta_1 \le \dots \le \beta_{b-1}=1$ are given parameters, $f:\R \to \R$ is the unknown. We show that there is a unique bounded function $f$ which solves (F) and satisfies $f(t)=0$ for $t<-1/(1-a)$, $f(t)=1$ 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.