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Let $f$ be absolutely continuous. Prove that $\hat{f}(n)=o\left(\frac{1}{n}\right)$.

Any hint will be appreciated, thanks.

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For $f$ absolutely continuous you have a representation $f(x) - f(0) =\int_0^x g(s) ds$ with some $g\in L^1$. – user20266 Dec 10 '11 at 15:58
up vote 1 down vote accepted


  1. Absolute continuous means that $f'\in L^1$. What can you say about the decay of $\hat{f'}$?

  2. Try to lift up the information from 1. to find the decay information on $\hat{f}$.

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