Find $f\in C^0(S^1)$ that satisfy $\limsup_{n\to \infty }\|S_nf -f\|_{L^\infty }>0$

Question

I have to construct a continuous function $f\in \mathcal C^0(S^1)$ (where $S^1=\mathbb R/\mathbb Z$) that satisfy $$\limsup_{n\to \infty }\|S_n f-f\|>0$$ where $S_nf$ is the $n-$th Fourier partial series. I absolutely don't know how to do (I in fact thought that it always converge to $0$). Any help would be welcome.

Answer 1

I bet that the problem does not ask you to "find" or construct" such an $f$. It bet it asks you to show that such an $f$ exists.

This is a standard application of the Uniform Boundedness Principle (aka the Banach-Steinhaus Theorem).

Hint: Define $T_n:C(S^1)\to\Bbb C$ by $$T_nf=S_nf(0).$$Show that $||T_n||=||D_n||_1$, where $D_n$ is the Dirichlet kernel. Hence $||T_n||\to\infty$...