Fourier asymptotics of Cantor type measures at infinity期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Fourier asymptotics of Cantor type measures at infinity

Authors:	Tian-You Hu Ka-Sing Lau

Institution:	Department of Mathematics, University of Wisconsin-Green Bay, Green Bay, Wisconsin 54311 ; Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

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 ; $\varlimsup_{t\to \infty }\phi (t)=\phi (\pi )$ if is odd and $\varlimsup_{t\to \infty}\phi (t)\le\phi (\pi )$ if 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.

Keywords:	Cantor type measure Fourier transform

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