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Fourier asymptotics of Cantor type measures at infinity


Authors: Tian-You Hu and Ka-Sing Lau
Journal: Proc. Amer. Math. Soc. 130 (2002), 2711-2717
MSC (2000): Primary 42A38; Secondary 26A12
DOI: https://doi.org/10.1090/S0002-9939-02-06398-0
Published electronically: April 17, 2002
MathSciNet review: 1900879
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Abstract: Let $q\geq 3$ be an integer and let $\phi (t)=\prod_{n=1}^\infty \cos(q^{-n}t)$. In this note we prove that $\lim_{t\to \infty} \phi (t)=-\phi (\pi )$ for all $q$; $\varlimsup_{t\to \infty }\phi (t)=\phi (\pi )$ if $q$ is odd and $\varlimsup_{t\to \infty}\phi (t)\le\phi (\pi )$ if $q$ is even$.$ This improves a classical result of Wiener and Wintner. We also give a necessary and sufficient condition for the product $\prod_{i=1}^m\phi (\alpha _it)$ to approach zero at infinity.


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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

Ka-Sing Lau
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
Email: kslau@math.cuhk.edu.hk

DOI: https://doi.org/10.1090/S0002-9939-02-06398-0
Keywords: Cantor type measure, Fourier transform.
Received by editor(s): February 4, 2001
Received by editor(s) in revised form: April 20, 2001
Published electronically: April 17, 2002
Additional Notes: Research supported by an HKRGC grant.
Communicated by: Andreas Seeger
Article copyright: © Copyright 2002 American Mathematical Society

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