Fourier asymptotics of Cantor type measures at infinity
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- by Tian-You Hu and Ka-Sing Lau PDF
- Proc. Amer. Math. Soc. 130 (2002), 2711-2717 Request permission
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.References
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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
- MR Author ID: 190087
- Email: kslau@math.cuhk.edu.hk
- 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
- © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1900879