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?
To show $\sum_{n=1}^{\infty}a_{n}\sin nx$ is a Fourier series of $f\in L^{\infty}$ if $na_{n}=O(1).$
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real-analysis
fourier-series
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0First, you can show with a Abel transform that the series is converging for all $x$. – 2012-05-01