If I have a function given as $f(x) = \exp(-|x|)$, is this function in the Schwartz space?
As a common example of the Schwartz function is considered $ e^{x^2}$. But in my example I got a absolute value, so the integral in $\Bbb R$ converges. The Schwartz function are functions such that they decrease rapidly in the infinity, so do they derivatives.
Well If I try to take a derivative from my given function, I got $\operatorname{sign}(x)*\exp(-|x|)$, and so on. But it doesn't seem to be a rapid decrease. Is there any general way how to test any function, if it is in Schwartz space?