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I want to show that if $a_{n}\downarrow 0$ and $na_{n}=O(1)$ then $\displaystyle \sum_{n=1}^{\infty}a_{n}\sin nx$ is a Fourier series of $f\in L^{\infty}$. Can anybody help me?

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First, you can show with a Abel transform that the series is converging for all $x$. –  Davide Giraudo May 1 '12 at 9:09

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