There is a classical theorem of Jackson stating that the $N$-th partial sum $S\_N f$ of the Fourier series of a Lipschitz continuous function $f$ (which is periodic with period 1) satisfies
$$
|f(x) - S\_N f(x)| \leq c \frac{K \log N}{N}
$$
uniformly for all $x \in [0,1]$, where $c$ is an absolute constant and $K$ is the constant in the Lipschitz condition. (This is stated in a more general case in the Wikipedia article here: <http://en.wikipedia.org/wiki/Convergence_of_Fourier_series#Uniform_convergence> )


Question: does anybody know where I can find a numeric value for the constant $c$? (It is particularly important for me that $c$ is independent of the function $f$ - this fact cannot be seen from the statement in Zygmund's book, for example). 


Be careful: Jackson's theorem is often stated in the form of the error between $f$ and the "best approximation" of $f$ - this is *not* the same as the error in the approximation by the partial sum of the Fourier series!


If you know where I can find the requested inequality, please let me know.



Jackson's theorem for partial sum of Fourier series
The estimate in the wikipedia article you linked does what you want because the modulus of continuity satisfies $\omega(2\pi/N)\le 2\pi L/N$ ($L$ = Lipschitz constant), but here's a derivation from scratch:


For convenience, let's focus on $x=0$, and assume that $f(0)=0$. We want to bound
$$
(S\_Nf)(0) = \int\_{-\pi}^{\pi} f(x) D\_N(x)\, dx ,
$$
where $D\_N(x)=\sin(N+1/2)x/\sin x/2$ is the Dirichlet kernel.


The part of the integral over $|x|\le 1/N$ is $\lesssim L/N$, with an absolute ($f$ independent) constant because $|f(x)/\sin x/2|\lesssim L$. To estimate the contributions coming from $|x|>1/N$, we integrate by parts, integrating $\sin(N+1/2)x$ and differentiating the rest. The boundary terms are $O(L/N)$ and
$$
\frac{1}{N+1/2}\left| \int\_{1/N\le|x|\le \pi} \left( \frac{f(x)}{\sin x/2}\right)'
\cos(N+1/2)x\, dx \right| \lesssim \frac{L\ln N}{N} ,
$$
because $|f'|\le L$ by assumption. (Also, when you differentiate $\sin x/2$ in the denominator, use that $|f(x)|\le L|x|$ in the estimate.)


