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Multiple spectra of Bernoulli convolutions. (English) Zbl 1365.28009

Let \(0< \lambda<1\) and let \(\mu_\lambda\) be the Bernoulli convolution associated with \(\lambda\). It is known that if \(\lambda = \frac{1}{2k}\) for some \(k\in \mathbb{N}\), then \(\mu_{\frac{1}{2k}}\) is a spectral measure with spectrum \(\Gamma(\frac{1}{2k})\) and that for certain odd integers \(p\) the multiple set \(p\,\Gamma(\frac{1}{2k})\) is also a spectrum for \(\mu_{\frac{1}{2k}}\). The authors find several conditions on \(p\) such that \(p\,\Gamma(\frac{1}{2k})\) is a spectrum for \(\mu_{\frac{1}{2k}}\).

MSC:

28A80 Fractals
42A65 Completeness of sets of functions in one variable harmonic analysis
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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