Solution in the distribution set

Question

Could you please help me to prove that the solution of

$$(1-\exp^{2i\pi x})T=0$$ in $D'(\mathbb{R})$ is $\sum_{n\in \mathbb{Z}}c_n\delta_n$ where $c_n=C$ for all $n\in \mathbb{Z}$

Answer 1

I am going to assume by $\delta_n$ you mean the shifted Dirac distribution. i.e. $i<\delta_n,\phi> = \phi(n)$.

Let $\phi \in \mathcal{D}(\mathbb{R})$ that is to say is smooth with compact support, then by definition we have that,

$\begin{align} <\sum_{n \in \mathbb{Z}}\delta_n(1-e^{2\pi i x}),\phi> &= \sum_{n \in \mathbb{Z}}<\delta_n, (1-e^{2\pi i x})\phi>\\ &= \sum_{n \in \mathbb{Z}}((1-e^{2\pi i n})\phi(n)\\ &=\sum_{n \in \mathbb{Z}}((1-1)\phi(n)=0 \end{align}$

Where the second to last equality follows from the fact that $e^{2\pi i n}=1$ for all $n \in \mathbb{Z}$.