Let $f$ be a Schwartz function in $\mathbb{R}^n$. I would like to know if there is some constant $\alpha > 0$ such that the function defined by $g(x) = f(x)e^{\frac{\lvert x \rvert^2}{\alpha}}$ is in the Schwartz space or even $L^2(\mathbb{R}^n)$.
Clearly this is true for Gaussian functions which are dense in the Schwartz space in some topology.
Best Answer
Let $f_0(x)= 1_{\{|x|\geq 1\}}(x)e^{-|x|}$ and let $\psi_1,\psi_2$ be a partition of unity subordinate to the cover $\{\{|x|>1\}, \{|x|<2\}\},$ with $\psi_1$ having support in the former set. Then, $f:= \psi_1f_0$ is a smooth function. It is easily seen to be a Schwartz function, and it also clearly doesn't satisfy your hypothesis, since
$$ |e^{\frac{|x|^2}{\alpha}}f(x)|\to\infty $$ as $|x|\to\infty$, no matter the value of $\alpha>0$.