Let $p(D)$ be a nontrivial polynomial with $D=-i\partial$ such that $$ p(D) = \sum_{a} c_aD^a $$ for a multiindex $a$ and constants $c_a$. I was wondering how and if we can prove that a fundamental solution $u^* \in \mathscr{D'}(\mathbb{R}^n)$ (in the space of distributions) given by $p(D)u^* = \delta$ exists.
I thought that we can start by approximating $\delta \in \mathscr{D}'$ by smooth, compactly supported functions since $\mathscr{D}\equiv C^\infty_c$ is dense in $\mathscr{D}'$. But then how do we show that this will converge in the limit?
Another possibility could be by Fourier analysis, but I am not sure how to progress with a proof. Any ideas?
EDIT
Eventually this holds as there is a theorem for it in Hörmander's book Analysis of Linear Partial Differential Operators I (see Theorem 7.3.10).
One of possible approaches is, indeed, to consider Fourier transform. After applying this transform you will get (up to some non-interesting normalisation constants) an equation $$p(\xi) \hat u = 1.$$
Now you need to show that (with some reasoning involving, eventually, principal values) $\frac{1}{p(\xi)}$ is a tempered distribution. Main points are that on infinity this fraction is bounded, and that near roots of $p$ the fraction "behaves well".
After that, you can apply an inverse FT to get your tempered distribution $u$.