Absolute continuity of Bernoulli convolutions, a simple proof. (English) Zbl 0867.28001
The authors give a new simplified proof that the infinite Bernoulli convolution \(\nu_\lambda\) of the measures
\[
\textstyle{{1\over 2}} (\delta_{-\lambda^n}+\delta_{\lambda^n})
\]
for \(n\in\mathbb{N}\) and \({1\over 2}<\lambda<1\) is absolutely continuous with \(L^2\)-density. The proof is based on differentiation techniques for measures.
Reviewer: H.Haase (Greifswald)
MSC:
28A12 | Contents, measures, outer measures, capacities |
42A61 | Probabilistic methods for one variable harmonic analysis |
28A15 | Abstract differentiation theory, differentiation of set functions |