# Order of partial sums in the derivatives of the Fourier series

Given periodic function $f\in C^{w}[0,1]$ with its Fourier series $f(x)=\sum\limits_{s=-\infty}^{\infty}f_{s}\exp(2\pi isx)$. What can one say about the asymptotic order of

$$\sum_{s=-K}^{K}\left(\frac{s}{K}\right)^{q}f_{s}\exp(2\pi isx)$$ for $q>w$ and $K\rightarrow\infty$? Is it just a constant?

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What is $w$? An integer? –  Davide Giraudo Sep 7 '12 at 21:54
Yes, w is integer, w>0 –  Katja Sep 8 '12 at 10:38
In the Fourier series it's $f_s$, not $f_i$. Do you want to compute the limit of the sum you wrote? –  Davide Giraudo Sep 8 '12 at 10:41