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:

$ f \in C^\infty (\mathbb R) $
$ \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?

Which norm is $|f|$? For the Schwartz space I remember the definition $\lim_{|x|\to\infty}|x^\alpha D^\beta f(x)|=0$. (super-polynomial decrease at infinity) insteat your assertion 2. — Mundron Schmidt, May 26 '17 at 08:09
I took it from my book. Anyway how do you prove that a function is S(R) using "your" definition? PS: I think should be an L1 norm — PeppeDAlterio, May 26 '17 at 08:32
It depends on your function. The Schwartz space is known as a space of very fast decreasing functions. The best example is $f(x)=e^{-|x|^2}$. Each derivative of $f$ is of the form $p\cdot f$ where $p$ is a polynom, even for $x^\alpha D^\beta f(x)$. Since the exponential term dominates the polynom, it is decreasing to $0$ at infinity. — Mundron Schmidt, May 26 '17 at 08:39
So everytime I have a function $C^{\infty} $ that is "faster" than a polynom (at the denominator) I can say that function is $ \mathscr S(R) $ ? — PeppeDAlterio, May 26 '17 at 11:04
Since $\alpha$ is arbitrary your function should decrease faster than each polynomial. — Mundron Schmidt, May 26 '17 at 11:15
What the second part of your definition should say is that for every choice of non-negative integers $\alpha$ and $\beta$, $$ |f|{\alpha,\beta} := \sup{x \in \mathbb{R}} \lvert x^\alpha D^\beta f(x) \rvert < +\infty. $$ — Branimir Ćaćić, May 26 '17 at 12:25
@MundronSchmidt and PeppeDAlterio The link in my post provide the proof of the equivalence between the definitions (as well as the proof that Schwartz functions are "rapidly decreasing functions"). — Pedro, May 31 '17 at 15:59

score 1 · Accepted Answer · edited Jun 12 '20 at 10:38

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.

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

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