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Riemann Lebesgue Lemma states that Fourier transform of an $L^1$ function, $\hat{f}(\lambda)$ is continuous and goes to zero as $|\lambda|\to \infty$. If $\mu$ is a finite nonatomic measure then is it true that $\hat{\mu}(\lambda)\to 0$? If not then is it true for some restricted class of finite measures? What are the restrictions?

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The following web site has a review article on work related to this question: http://mypage.iu.edu/~rdlyons/pdf/seventy.pdf

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  • $\begingroup$ Comment just for those skimming this page without reading the link: the established jargon for such a measure is a Rajchman measure $\endgroup$
    – Yemon Choi
    Commented Mar 30, 2013 at 22:55

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