# Asymptotic order of some sums with the Fourier coefficients

Given $f\in C^{w}[0,1]$ with periodic conditions $f(0)^{(j)}=f(1)^{(j)},\ j=0,\dots, w-1$ and its Fourier series are $f(x)=\sum_{l}f_{i}\exp(2\pi ix)$.

I need to find the asymptotic order of errors $$\sum_{\left|l\right|>K/2}f_{l}\exp(2\pi i lx)\mbox{ and }K^{-4q}\sum_{l=0}^{K/2}f_{l-K}l{}^{4q}\exp\left(2\pi i lt\right)\mbox{ as }K\rightarrow \infty.$$

Is it enough to say that $\left|f_{i}\right|=O(K^{-w})$ (from integration by parts) and replace the summation by integration to show that this order is $O(K^{1-w})$?

Or when the orders of corresponding sum and integral are the same?

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