Uniform convergence of Fourier series in terms of modulus of continuity

Question

In the section Uniform Convergence in https://en.wikipedia.org/wiki/Convergence_of_Fourier_series it is given the result of D. Jackson which states that if $f \in C^p$ is $2\pi$-periodic and $f^{(p)}$ has modulus of continuity $\omega$, then $$ |f(x) - (S_N f)(x)| \leq K \dfrac{\log N}{N^p}\omega(2\pi/N) $$ for $K > 0$ a constant independent of $N,f,p$. Here $S_N f$ denotes the $N$-th partial sum of the Fourier series expansion of $f$. I have been unable to find the proof of this and was wondering if anyone would be kind enough to show it? Thanks!

Answer 1

I guess this is the page of Jackson’s book referenced by Wikipedia's article.