I want to show that if $\displaystyle\sum a_{n}\sin nx (a_{n}\downarrow 0)$ is a Fourier series of $f\in L^{1}$ then $\displaystyle \sum \frac{a_{n}}{n}<+\infty.$ I know i have to use some property of Dirichlet's kernel but i am stuck how to use them to derive my result.
Let $a_{n}\downarrow 0$ and if series $\sum a_{n}\sin nx$ is a Fourier series of function $f\in L^{1}$ then $\sum \frac{a_{n}}{n}<+\infty.$
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real-analysis
fourier-series
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0See the comments to [this question](http://math.stackexchange.com/questions/129575/series-which-are-not-fourier-series). – 2012-04-18
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0@JuliánAguirre The comments in that question assume $f\in L^1([-\pi,\pi]) \subset L^2([-\pi,\pi]),$ a stronger condition than what I assumed. – 2012-04-18
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0@RagibZaman You assumed $f\in L^2$, which is stronger than $f\in L^1$. – 2012-04-18
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0@JuliánAguirre My bad, I see now. – 2012-04-18