Let $f\in\mathcal{S}(\mathbb{R}^n),$ the Schwartz class. Is it true that the function $g$ defined on $\mathbb{R}_+=[0,\infty),$ by $$g(r)=\int_{S^{n-1}} f(rw)d\sigma(w) $$ also in Schwartz class $\mathcal{S}(\mathbb{R_+})?$
($d\sigma$ is the normalised surface measure of $S^{n-1}$)