Can a schwartz class function be dominated by an exponential?

Question

Given a function $f$ from Schwartz class, does there exist a constant $C$ such that $|f(x)|<Ce^{-|x|}$. For me its seems true, if it is not true, any counterexample would be very illustrative to me

Answer 1

A simple counterexample is $f(x)=e^{-\sqrt{|x|}}$ (OK, that's not smooth at $0$, but just smooth it out near $0$ since all we care about is the behavior for $x$ large).

Much more generally, suppose $f_0,f_1,f_2,\dots$ are any countable collection of Schwartz class functions. Then we can find a Schwartz class function $f$ such that for all $n$ there exists $x$ such that $|f(x)|>|f_n(x)|$. The proof is fairly complicated and involves a diagonal argument: piece by piece you define $f$ on larger and larger intervals, so that you alternate between adding the countably many constraints that will make $f$ Schwartz and making $f$ bigger than each $f_n$ somewhere.