  Using a version of Hironaka's resolution of singularities for real-analytic
functions, any elliptic multiplier $\mathrm{Op}(p)$ of order $d>0$,
real-analytic near $p^{-1}(0)$, has a fundamental solution $\mu_0$. We give an
integral representation of $\mu_0$ in terms of the resolutions supplied by
Hironaka's theorem. This $\mu_0$ is weakly approximated in
$H^t_{\mathrm{loc}}(\mathbb{R}^n)$ for $t<d-\frac{n}{2}$ by a sequence from a
Paley-Wiener space. In special cases of global symmetry, the obtained integral
representation can be made fully explicit, and we use this to compute
fundamental solutions for two non-polynomial symbols.
