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Asymptotic behavior of Fourier transforms of self-similar measures


Author: Tian-You Hu
Journal: Proc. Amer. Math. Soc. 129 (2001), 1713-1720
MSC (2000): Primary 42A38; Secondary 28A80.
DOI: https://doi.org/10.1090/S0002-9939-00-05709-9
Published electronically: November 3, 2000
MathSciNet review: 1814101
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Abstract:

Let $\mu $ be a self-similar probability measure on $\mathbb{R}$ satisfying $ \mu =\sum_{j=1}^mp_j\mu \circ F_j^{-1},$ where $F_j(x)=\rho x+a_j,$ $0<$ $ \rho <1,$ $a_j\in \mathbb{R},$ $p_j>0$ and $\sum_{j=1}^mp_j=1.$ Let $\hat{\mu} (t)$ be the Fourier transform of $\mu .$ A necessary and sufficient condition for $\hat{\mu}(t)$ to approach zero at infinity is given. In particular, if $a_j=j$ and $p_j=1/m$ for $j=1,...,m,$ then $\lim \sup_{t\rightarrow \infty }\vert\hat{\mu}(t)\vert>0$ if and only if $\rho ^{-1}$ is a PV-number and $\rho ^{-1}$ is not a factor of $m$. This generalizes the corresponding theorem of Erdös and Salem for the case $m=2.$


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Additional Information

Tian-You Hu
Affiliation: Department of Mathematics, University of Wisconsin-Green Bay, Green Bay, Wisconsin 54311
Email: hut@uwgb.edu

DOI: https://doi.org/10.1090/S0002-9939-00-05709-9
Keywords: Fourier transform, PV-number, self-similar measure
Received by editor(s): August 6, 1999
Received by editor(s) in revised form: September 16, 1999
Published electronically: November 3, 2000
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2000 American Mathematical Society

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