The gaussian function $\mathrm{e}^{-|\xi|^2}$ is a schwartz function. In general, Are there some critery on the function $f(\xi)$ for which the function $\mathrm{e}^{-f(\xi)}$ is a Schwartz funtion?
From what I see, $f(\xi)=|\xi|^{m}$ with $m$ any even integer seems to work and I think $f(\xi)=a_m|\xi|^m+a_{m-1}|\xi|^{m-1}+\cdots+a_1|\xi|+a_0$ with $a_m>0$ and $m$ even integer it does too.