Hi there I have got a piecewise continuous function $f(x)$ that decays zero exponentially for x large.
Can I approximate it by a differentiable function quantitatively? "like
$
\big|f(x)-L(x) \big| \ll \frac{1}{x\log x} "
$
What is the best I can do?
It seems that the answer is positive even for much more general case of an (open) subset $U$ of separable real Banach space admitting a separating polynomial (in particular, for $U=\mathbb R$) and analytic approximation (see Theorem 2 here). But if you are interested only in your particular case then probably somewhere is a much more simple proof of it.