Convergence of Fourier series for $C^p$ functions

Question

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). $

Answer 1

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.

Answer 2

Well, not so simple, you should study the well-written MR0067227 (16,692a) Satô, Masaka Uniform convergence of Fourier series. Proc. Japan Acad. 30, (1954). 528–531.