As we know, for $1<p<\infty$, the Fourier series of $f\in L^{p}(T)$ converges to $f$ in $L^{p}$-norm.
But is there any results concerning the convergence of Fourier series in $L^{\infty}$-norm?
Since $L^{\infty}(T)$ is not separable, the trigonometric system fails to form a Schauder basis of $L^{\infty}(T)$, this implies that the Fourier series of $L^{\infty}(T)$-functions fails to converge in $L^{\infty}$-norm. But does the Fourier series of $f$ converge in $L^{\infty}$-norm for every $f\in C(T)$?



Convergence of Fourier series in L^{\infty}-norm
If Fourier series of continuous functions would converge in $L^\infty$, then, by the Uniform Boundedness Principle, the operator norms in $C(\mathbb{T})$ of the partial Fourier series operators $S\_Nf(t):=\sum\_{n=-N}^N\hat{f}(n)e^{int}$ would be uniformly bounded. You can find, for example in Katznelson book, a proof of the fact that such norms diverge logarithmically.


