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March 31st, 2018, 02:24 PM   #1
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From: DK

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Distribution - Fourier series

Good day, I am trying to solve an exercise in the course of distribution theory and fourier analysis. I am new to the matter of using distribution in calculating, and I am thankful for any help to solve the following question:

1. Consider the $2\pi$ -periodic function $f(x)$ defined on $[0,2\pi)$ by $f(x)=\frac{1}{2}(\pi-x)$.
prove (by calculating the Fourier series) that in the sense of distributions
$\sum_{n=1}^{\infty }\frac{sin(nx)}{n}=f(x)$.

2. prove that-in the sense of distributions

$\sum_{n\in \mathbb{Z}}e^{inx}=2\pi\sum_{n\in \mathbb{Z}}\delta_{2\pi n}$ in $D^{'}(\mathbb{R})$
Where $\delta_{2\pi n}$ is the distribution $\phi \mapsto \phi (2 \pi n)$

Thanks.
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