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The partial sum of the Fourier series of $f(t)=(t-\pi)\chi_{\left(0,2\pi\right)}$ can be written as \begin{align} S_n(f,t) &= \sum_{0<|k| \le n} \frac{i}{k} e^{ikt} \; (1) \\ & = -2\sum_{k=1}^n \frac{\sin kt }{k} \; (2)\\ &= 2 \int_0^t \left( \frac{\sin (n+1/2)t}{\sin t/2} - 1 \right) dx \; (3) \end{align}

and it can be shown that $ \left \lvert S_n(f,t) \right \rvert \le \pi+2 $ Define $g_n(t)=\frac{e^{int}}{\pi+2} S_n(f,t)$, show that for some $c>0$ $ \left \lvert S_n(g_n,0) \right \rvert > c \ln n $ where $S_n(g_n,0)$ is the partial sum of the Fourier series of $g_n$ evaluated at $0$.

I found that the partial sum of the Fourier series of $g_n$ is $ (\pi+2)S_n(g_n,t)=\sum_{0\le k \le 2n, k\ne n} \frac{i}{k-n}e^{ikt} $ Evaluated at zero, this looks like a taylor series of $\ln(x-1)$, but this won't get me near the result.
I know the Dirichlet kernel as a $L^1$ norm proportional to $\ln n$.
But I don't see how to use (3) to use this fact.

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    @joriki a) You are right. b) I agree. c) I see your point, but I can't find my error. Writing $\hat f_n$ for the $n$-th Fourier coefficient of $f$, I have $\hat f_0=0$ and $\hat f_n=\frac{1}{2 \pi}\int t e^{-int}dt=\frac{1}{2 \pi}\frac{te^{-int}}{(-in)}+0$.2012-10-15

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