Functions for which all moments are well-defined and finite

Question

I understand Schwartz functions (informally) as the set of functions $f$ such that

1. $f \in C^\infty$,
2. $f$ decays at $\pm \infty$ faster than any polynomial,
3. $\frac{d^nf}{dx^n}$ decays at $\pm \infty$ faster than any polynomial, for all $n = 1, 2, ...$.

I am wondering: is there a name for the class of functions which satisfy only property 2?

Really what I'm after is the class of functions for which every moment is well-defined and finite.