Translator Disclaimer
1 November 2003 Spectra of Bernoulli convolutions as multipliers in $L^p$ on the circle
Nikita Sidorov, Boris Solomyak
Duke Math. J. 120(2): 353-370 (1 November 2003). DOI: 10.1215/S0012-7094-03-12025-6

Abstract

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 R. Salem and P. Sarnak, this proves that for every fixed $θ > 1$ the spectrum of the convolution operator f μ θ f in $L^p(S^1)$ (where $S^1$ is the circle group) is countable and is the same for all p ( 1 , ) , namely, { μ θ ^ ( n ) : n } ¯ . Our result answers the question raised by Sarnak in [8]. We also consider the sets { μ θ ^ ( r n ) : n } ¯ 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 ( θ ) but uncountable (a nonempty interval) for Lebesgue-a.e. $r>0$.

Citation

Download Citation

Nikita Sidorov. Boris Solomyak. "Spectra of Bernoulli convolutions as multipliers in $L^p$ on the circle." Duke Math. J. 120 (2) 353 - 370, 1 November 2003. https://doi.org/10.1215/S0012-7094-03-12025-6

Information

Published: 1 November 2003
First available in Project Euclid: 16 April 2004

zbMATH: 1059.47004
MathSciNet: MR2019979
Digital Object Identifier: 10.1215/S0012-7094-03-12025-6

Subjects:
Primary: 47A10
Secondary: 11R06, 42A16

Rights: Copyright © 2003 Duke University Press

JOURNAL ARTICLE
18 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.120 • No. 2 • 1 November 2003
Back to Top