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.