Exercise 12, § 4

For $t > 0$ and $x \in \mathbf{R}$, let $F_t(x) = te^{-\pi t^2 x^2}$. Let $f \in \overline{\mathscr{K}(\mathbf{R})}$. Show that $f * F_t$ tends to $f$ uniformly as $t$ tends to $+\infty$. (Show that the measures $F_t(x) dx$ satisfy the conditions of §2, No. 7, Lemma 4, with $a = 0 .$)