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March 31st, 2018, 01:24 PM  #1 
Newbie Joined: Mar 2018 From: DK Posts: 1 Thanks: 0  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 $f(x)=\frac{1}{2}(\pix)$. prove (by calculating the Fourier series) that in the sense of distributions $\sum_{n=1}^{\infty }\frac{sin(nx)}{n}=f(x)$. 2. prove thatin 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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distribution, fourier, series 
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