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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 $g\in L^1$.2011-12-10

1 Answers 1

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Hint:

  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 $\hat{f}$.