A function $u:\mathbb{R}^n\rightarrow \mathbb{C}$ belongs to the Schwartz class $\mathcal{S}(\mathbb{R}^n)$ if \begin{equation*} \sup_{x\in \mathbb{R}^n} |x^{\alpha} \partial^{\beta} u |<\infty \end{equation*} for all multi-indices $\alpha,\beta\in \mathbb{Z}^{+}$. Typical examples are the Gaussian $x\mapsto e^{-a |x|^2}$, $a>0$, and smooth functions with compact support traditionally denoted by $C^{\infty}_{c}(\mathbb{R}^n)$.
I wonder if there are more known explicit examples of functions in $\mathcal{S}(\mathbb{R}^n)$.
In fact, I wonder if there are explicit examples for functions in $C^{\infty}_{c}(\mathbb{R}^n)$ other than the mollifiers generated by the smooth function $t\mapsto e^{-\frac{1}{t}}$.
Of course, if we mollify any $L^{p}$ function, $p\geq 1$, we obtain an approximating sequence of functions in $C^{\infty}_{c}(\mathbb{R}^n)$. But could any one give more explicit examples ?