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I'm dealing with the problem, that the function defined by trigonometric series(that is, limit of symmetric sum of $e^{inx}$)

I have shown that this function converges everywhere in pointwise sense. However, since it is not uniformly converge, I cannot say that this series is indeed a Fourier series of the give function. Can I say more about the relation between Fourier series of it and itself??

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There are know examples of trigonometric series that are no Fourier series. Do we talk about a specific series or is it a general question? –  AD. Oct 13 '12 at 11:21
An example of a trigonometric series that is not a Fourier series is given by $$\sum_{n>0}\frac{\sin nx}{\log n}$$ see my answer on this –  AD. Oct 13 '12 at 11:26

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