An equation with distributions

Question

I am trying to solve the equation $\sin(x) T = 0$ where $T \in \mathcal{D}'(\mathbb{R})$ is a distribution.

I know that for such a distribution we have $\textrm{supp}(T) \subset \pi \mathbb{Z}$, but I don't know how to move on from here ...

Thank you for your help.

Answer 1

Take $\phi \in C^\infty_c(-1,1),\phi(0)=1$. For all $\psi\in C^\infty_c$ $$\langle T,\psi \rangle = \langle T \sin,\frac{\psi-\sum_n \psi(\pi n)\phi(.+\pi n)}{\sin} \rangle + \sum_n \psi(\pi n)\langle T,\phi(.+\pi n)\rangle$$

$$ = \sum_n \langle T,\phi(.+\pi n)\rangle\langle \delta(.-\pi n),\psi\rangle$$ $$ = \langle \sum_n \langle T,\phi(.+\pi n)\rangle\delta(.-\pi n),\psi\rangle$$

Answer 2

You have simply $$ T(x) = \sum_{k\in\mathbb{Z}}\lambda_k\delta(x-k\pi), \;\lambda_k\in\mathbb{R} $$

N.B. : $T(x)$ doesn't contain derivatives of Dirac deltas, because $x^n\delta^{(m)}(x)\neq0\;\,\forall n \le m$ and $\sin(x) \sim x$.