I am having a bit of a confusion regarding convergence results.
Suppose $f$ is Lipschitz, or $f \in C^\infty$ and let $S_{N}f$ be its truncated Fourier series. In the wikipedia page http://en.wikipedia.org/wiki/Convergence_of_Fourier_series, the following is stated:
$|f(x)-S_{N}f(x)| \le K \frac{\log N}{N}$
where $K$ is independent of $f$. In Katznelson's "An Introduction to Harmonic Analysis", a similar result is given as an exercise (first exercise of Chapter 3), however $K$ there strongly depends on $f$ (it is the Lipschitz constant).
My question is - is there a theorem regarding an error term that decays w.r.t. $N$ that does not depend on $f$ itself?
Thank you very much.