Check if a function is $\mathscr S(\mathbb R)$

Question

How can I check if a generic function is $ \mathscr S(\mathbb R) $ ? I mean the Schwartz space.

The definition asserts that $ f\in \mathscr S(\mathbb R) $ if:

1. $ f \in C^\infty (\mathbb R) $
2. $ \displaystyle\Vert f \Vert = \sup_{x\in\mathbb R} | x^\alpha D^\beta f(x) | $

Is there an easy way to check the assertion 2?

Answer 1

Proposition. Let $f\in C^\infty(\mathbb R^n)$. The following statements are equivalent.

(a) $\displaystyle\sup_{x\in\mathbb{R}^n}|x^\alpha D^\beta f(x)|<\infty$ for all $\alpha,\beta\in\mathbb{N}^n$.

(b) $\displaystyle\sup_{x\in\mathbb{R}^n}\|x\|^k|D^\beta f(x)|<\infty$ for all $k\in \mathbb N$ and $\beta\in\mathbb{N}^n$.

(c) $\displaystyle\lim_{|x|\to\infty}\|x\|^kD^\beta f(x)=0$ for all $k\in \mathbb N$ and $\beta\in\mathbb{N}^n$.

(d) $\displaystyle\lim_{|x|\to\infty} x^\alpha D^\beta f(x)=0$ for all $\alpha,\beta\in\mathbb{N}^n$.

Proof: see here.

So, to check the assertion 2 you can check any of the assertions (b), (c) or (d). As there are some alternatives dealing with limits instead of supremum, the work can become easier.