Show that $\langle T, \varphi \rangle = \sum_{n=1}^{\infty} \varphi(n)$ is a tempered distribution on $\mathbf{R}^1$.
My question is from Strichartz book, A guide to Distribution theory and Fourier transforms. How do I show this? I'm thinking about using the Structure theorem, but not really sure on how to proceed with it.
Firstly the sum converges:
Due to the Equivalent definition of the Schwarz function, (I assume $\varphi$ is a Schwarz function although you didn’t say) there exist a constant $C>0$ such that $|\varphi(x)|\le C{(1+x^2)}^{-1}, \forall x\in \mathbb{R}$, thus $T(\varphi)\le \sum_{n\ge 1}C{(1+x^2)}^{-1}< \infty.$
Then we only prove continuity, linear is obviously by above.
Let $\varphi_{k}(x)\rightarrow 0$ in norm of Schwarz class, then by definition $$\sup_{x\in \mathbb{R}}|(1+x^2)\varphi_{k}(x)|\rightarrow 0.$$ thus $$|T(\varphi_{k})|=|\sum_{n\ge 1}\varphi_{k}(n)(1+n^2){(1+n^2)}^{-1}| \le \sup_{x\in \mathbb{R}}|\varphi_{k}(x)(1+x^2)|\sum_{n\ge 1}{(1+n^2)}^{-1} \rightarrow 0.$$