Let $f \in C^p[0,2\pi]$ and periodic. Denote $\omega\_p$ as the moduli of continuous of $f^{(p)}$. Then
$
|f - S\_Nf| \le K \frac{\log{N}}{N^p}\omega\_p(2\pi/N),
$
where $S\_N$ is the Fourier partial sum of order $N$, and $K$ is some constant. 


I could not prove this result. Can someone help me please? 


I found a similar result for the best approximation of trigonometric polynomial of order $N$, $f^\*$, 


of $f$ in 'An introduction to the approximation of functions' by Theodore J. Rivlin that
$
|f - f^\*| \le \frac{K}{N^p}\omega\_p(2\pi/N).
$



Convergence of Fourier series for $C^p$ functions
There is a theorem of Lebesgue that says that for a continuous periodic $f$,
$$
\|f - S\_N f\|\_\infty \le C \log N \|f - f^\* \|\_\infty.
$$
This appears as Theorem 2.2 in Rivlin's book. Combined with the result you already know, you get what you want. 


