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Example: for function $$f(x)=x^{3}(1-x)^{3}=\sum f_{s}\exp(2\pi isx)$$ Fourier series of its fourth derivative are different from derivative of its Fourier series $$f^{(4)}(x)=-360x^{2}+360x-72=\sum g_{s}\exp(2\pi i s x)$$ with $g_{s}\neq(2\pi s)^{4}f_{s}$

Related question: When $$\sum\left|f_{s}\right|^{2}j^{2p}<\infty$$ is equivalent to $f\in C^{(p)}[0,1]$? (for periodic f)

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    notice that the 3rd derivative of your function (when extended from $[0,2\pi)$ periodically to $\mathbb{R}$) is not continuous. Fourier series will correctly give its *distributional* derivative, which includes $\delta$-functions coming from jumps. You would need to subtract (the right multiple of) the Fourier series of $\delta$, which is just $\sum_k e^{2\pi i k x}$. For your second question, the inequality means that $f^{(p)}$ is in $L^2$, so at least $f\in C^{(p-1)}(S^1)$.2011-03-31
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    Thank you very much! I didn't notice it!2011-03-31
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    maybe you could have a look at the related question? http://math.stackexchange.com/questions/30744/one-more-question-about-decay-of-fourier-coefficients2011-04-04

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