Convergence of distributional approximate identities

Let the function $g \in L_{1}(\mathbb {R})$ and $\int_{\mathbb {R}} g d x = 1$. Let $F_{n} \in \mathcal{D}^{\prime }$ be the regular generalized functions generated by the functions $n g(n x)$. Prove that $F_{n} * \varphi \rightarrow \varphi$ in $\mathcal{D}$ for every function $\varphi \in \mathcal{D}$ (cf. problems 16.20 and 18.31 а)).

Suppose $\left\{ f_{i}\right\}$ is a sequence of locally integrable functions in $\mathbb {R}^{n}$, such that

exists, for each $\phi \in \mathscr {D}\left(\mathbb {R}^{n}\right)$ and each $x \in \mathbb {R}^{n}$. Prove that then $\left\{ D^{a}\left(f_{i} * \phi \right)\right\}$ converges uniformly on compact sets, for each multi-index $\alpha$.

If $\left\{ h_{j}\right\}$ is an approximate identity, as in Definition 6.31, and $u \in \mathscr {S}_{n}^{\prime }$, does it follow that $u * h_{j} \rightarrow u$ as $j \rightarrow \infty$, in the weak*-topology of $\mathscr {S}_{n}^{\prime }$?