# Distributions with compact support are tempered

1. Oct 18, 2011

### e(ho0n3

1. The problem statement, all variables and given/known data
Let $F \in \mathcal E'(\mathbf R)$. Prove that $F \in \mathcal S'(\mathbf R)$.

2. The attempt at a solution
Since $F \in \mathcal E'(\mathbf R)$, there exists a continuous function $f \colon \mathbf R \to \mathbf R$ and a nonnegative integer $k$ such that for every $\varphi \in \mathcal E(\mathbf R)$,

$$\langle F, \varphi \rangle = \int_{-\infty}^\infty f(x) \varphi^{(k)}(x) \, dx$$

To prove that $F \in \mathcal S'(\mathbf R)$, I need to prove that if $\varphi_n \to 0$ in $\mathcal S(\mathbf R)$, then $\langle F, \varphi_n \rangle \to 0$. Using the above integral representation for $F$, I was hoping to pass the limit under the integral sign using the dominated convergence theorem but I cannot think of any dominating integrable function. I don't know what else to do here. Any tips?