Density of $\mathscr{S}$ in $H^s(\mathbb{R}^n)$

Question

As usual, we define $H^s(\mathbb{R}^n)$ as the set of distributions $f\in\mathscr{S}'$ such that $\hat{f}\in L^2_{\text{loc}}$, and $$ \|f\|_{H^s}:=\left(\int \left(1+\left|\xi\right|^2\right)^s\left|\hat{f}(\xi)\right|^2\mathop{dx}\right)^{1/2} <\infty$$

How can I prove that $\mathscr{S}$ is dense in $H^s$ for all $s\in \mathbb{R}$?

Answer 1

Taking the Fourier transform it suffices to show that $\widehat{\mathscr{S}}=\mathscr{S}$ is dense in $L^2(\Bbb{R}, (1+|x|)^{2s}dx)$ which follows from that $(f1_{|f|<A})\ast ne^{-\pi n^2 x^2}$ is Schwartz, and $f1_{|f|<A}\xrightarrow{A\to\infty} f$, and $ne^{-\pi n^2 x^2}\xrightarrow{n\to\infty} \delta_0$.