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). $