# Convergence of Schwartz functions and tempered distributions

Suppose that $$\psi \in \mathscr{S}(\mathbb{R})^{*}$$ is a tempered distribution and $$\phi_n \in \mathscr{S}(\mathbb{R})$$ are Schwartz functions so that $$\phi_n(x) \rightarrow 1$$ pointwise. For example $$\phi_n(x) = e^{-x^2/n}$$. Denote by $$c$$ any constant function, e.g. $$c(x)=1$$.

My questions:

1) What would be conditions that assure that $$\langle \psi,c\rangle$$ makes sense (exists)? I mean beyond the usual integral conditions for specific $$\psi$$.

2) Under which conditions does it hold that $$\langle\psi, \phi_n\rangle \rightarrow \langle \psi,c\rangle$$ as $$n \rightarrow \infty$$? Or more generally how to approximate a function $$f$$ not in $$\mathscr{S}(\mathbb{R})$$ so that $$\langle\psi, \phi_n\rangle \rightarrow \langle \psi,f\rangle$$?

My very preliminary thoughts:

Obviously the convergence in 2) is not in $$\mathscr{S}(\mathbb{R})$$ since the constant function is not Schwartz. So the continuity of $$\psi$$ in $$\mathscr{S}(\mathbb{R})$$ does not imply 2). I have not found a question like this treated anywhere. Maybe I can use Hermite expansion to show that $$\langle\psi,\phi_n\rangle=\sum_m \langle\psi,p_m\rangle \langle p_m,\phi_n\rangle \rightarrow \sum_m \langle\psi,p_m\rangle \langle p_m,c\rangle$$ where $$p_m$$ are the Hermite functions. But even then how to show that the last term equals $$\langle\psi,c\rangle$$?

Thank you for ideas, comments, suggestions! I don't really know how to approach this.

Let $$T$$ be a tempered distribution. For a Schwartz function $$\varphi$$ then $$T \ast \varphi \in C^\infty$$.

If for some $$\varphi$$ Schwartz satisfying $$\int_{-\infty}^\infty \varphi(x)dx = 1$$ we have $$T \ast \varphi \in L^1$$ then let $$\langle T,1 \rangle = C, \qquad\qquad C = \int_{-\infty}^\infty T\ast \varphi(x)dx$$

$$\langle T,1 \rangle = C$$ makes sense in the sense that the value doesn't depend on the chosen $$\varphi$$.

Proof : the condition ensures that the tempered distribution $$\widehat{T}$$ is (represented by) a continuous function when restricted to $$C^\infty_c(-r,r)$$ with $$r$$ small enough such that $$\widehat{\varphi}$$ doesn't vanish there and $$C = \widehat{T}(0)$$.