Spectra of Bernoulli convolutions as multipliers in $L^p$ on the circle
| dc.creator | Sidorov, Nikita | |
| dc.creator | Solomyak, Boris | |
| dc.date | 2002-10-03 | |
| dc.date.accessioned | 2026-07-07T04:51:37Z | |
| dc.date.available | 2026-07-07T04:51:37Z | |
| dc.description | It is shown that the closure of the set of Fourier coefficients of the Bernoulli convolution $μ_θ$ parameterized by a Pisot number $θ$, is countable. Combined with results of Salem and Sarnak, this proves that for every fixed $θ>1$ the spectrum of the convolution operator $f\mapsto μ_θ*f$ in $L^p(S^1)$ (where $S^1$ is the circle group) is countable and is the same for all $p\in(1,\infty)$, namely, $\bar{\{\hat{μ_θ}(n) : n\in\mathbb{Z}\}}$. Our result answers the question raised by P. Sarnak in \cite{Sar}. We also consider the sets $\bar{\{\hat{μ_θ}(rn) : n\in\mathbb{Z}\}}$ for $r>0$ which correspond to a linear change of variable for the measure. We show that such a set is still countable for all $r\in\Q(θ)$ but uncountable (a non-empty interval) for Lebesgue-a.e. $r>0$. | |
| dc.description | 18 pages | |
| dc.identifier | https://arxiv.org/abs/math/0210053 | |
| dc.identifier | http://arxiv.org/abs/math/0210053 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/65167 | |
| dc.subject | Functional Analysis | |
| dc.subject | Dynamical Systems | |
| dc.subject | Number Theory | |
| dc.subject | 47A10; 42A16; 11R06 | |
| dc.title | Spectra of Bernoulli convolutions as multipliers in $L^p$ on the circle | |
| dc.type | text |