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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.

user301068
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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 $||T_n||\to\infty$...

David C. Ullrich
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  • No no, they ask me exactly to construct such a function. A proposition gives me already the existence. – user301068 Mar 22 '16 at 16:26
  • Well then good luck with that... After the problem's been handed in and handed back tell us what the answer was supposed to be - people will be curious. – David C. Ullrich Mar 22 '16 at 16:38
  • You have a famous example form Fejer, but as mentionned by David... good luck to engineer such an example by yourself! Fejer example is described in "Fourier series - A modern introduction" from R.E. Edwards. – mathcounterexamples.net Mar 22 '16 at 17:06