  Let $\mu$ be a Borel probability measure on $\mathrm{SL}_2(\mathbb R)$ with a
finite exponential moment, and assume that the subgroup $\Gamma_{\mu}$
generated by the support of $\mu$ is Zariski dense. Let $\nu$ be the unique
$\mu-$stationary measure on $\mathbb P^1_{\mathbb R}$. We prove that the
Fourier coefficients $\widehat{\nu}(k)$ of $\nu$ converge to $0$ as $|k|$ tends
to infinity. Our proof relies on a generalized renewal theorem for the Cartan
projection.
