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I have the next questions that I am stuck in them.

  1. Let $f = 0 $ for $x\in [0,\pi]$ and $f=1$ for $x\in [\pi,2\pi]$.

I need to find some constant $c$ such that for every natural N:

$$f*D_N \left(\pi+\frac{\pi}{N+1/2} \right)> 1+c \tag{$\dagger$}$$

($D_N$ is the Dirichlet kernel). 2. show that there exists a constant $c$ s.t $\forall N \in \mathbb{N}$, $\sum_{n=1}^{N} \frac{\sin(nt)}{t}<c$.

I see that assignment 2. is a consequence of 1 but I don't seem to prove either of them.

For 1. I have:

$$ \begin{align} f*D_N \left(\pi+\frac{\pi}{N+1/2} \right) &= \int_{\pi}^{2\pi} \frac{\sin((N+1/2)\pi+\pi-(N+1/2)y)}{\sin(1/2 \pi +\pi/(2N+1)-y/2)} \\ & = \int_{\pi}^{2\pi} \frac{\sin((N+1/2)(y-\pi))}{\cos(y/2-\pi/(2N+1))}\frac{dy}{2\pi} \end{align}$$

How to estimate this integral from below?

Thanks.

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