The problem is as stated in the title but in more detail: find all tempered distributions $U\in\mathcal{S'(\mathbb{R})}$ that solve the convolution equation given in the title. My approach uses the Fourier transform and a transform table. I know that $$ \mathfrak{F}(U) \cdot\mathfrak{F}(e^{-|x|})=\mathfrak{F}({\sin(2x)}), $$ with my table and rearranging I find $$ \mathfrak{F}(U)=\frac{\pi}{i}\big(\delta(\omega-2)-\delta(\omega+2)\big) \frac{2}{1+\omega^2}.$$ Then I would find $U$ by computing the inverse transform but I am having difficulties doing that with my table. Is this a good approach or is there a smarter way to do this. You find the table below.
