When does a Bernoulli convolution admit a spectrum? (English) Zbl 1266.42012
A discrete subset \(\Lambda\subset\mathbb{C}\) is a spectrum of a probability measure \(\mu\) if \(\{\exp(-2\pi i\lambda):\lambda\in \Lambda\}\) creates an orthogonal basis for \(L^2(\mu)\). A spectral measure is a probability measure that admits a spectrum. It is proved that the distribution of
\[
\sum^\infty_{n=1} \varepsilon_n\rho^n/2,
\]
called Bernoulli convolution, where \(\varepsilon_n\in \{-1,+1\}\) are choosen independently with probability \(1/2\), is not a spectral measure for irrational contraction rates \(\rho\) and that in the case of rational \(\rho\) the Bernoulli convolution fails to be a spectral measure unless \(\rho\) is the reciprocal of an even integer.
Reviewer: Wlodzimierz Ślȩzak (Bydgoszcz)
MSC:
| 42A65 | Completeness of sets of functions in one variable harmonic analysis |
| 42B05 | Fourier series and coefficients in several variables |
| 42C30 | Completeness of sets of functions in nontrigonometric harmonic analysis |
| 28A78 | Hausdorff and packing measures |
| 28A80 | Fractals |
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