Rapidly Decreasing Functions

Question

Can someone explain the notion of a rapidly decreasing function? Namely, a function in the Schwartz space:

$$\mathscr{S}(\mathbb{R}^n):= \{ f \in C^{\infty} (\mathbb{R}^n) : ||f||_{\alpha, \beta} < \infty \, \, \forall \alpha, \beta\}$$

with $$ ||f||_{\alpha,\beta}:= \sup_{x \in \mathbb{R}^n} |x^{\alpha} D^{\beta} f(x)|$$

What exactly is this trying to say? I'm not quite familiar with the notation $x^\alpha$. Is this the same $x \in \mathbb{R}^n$ that we take the sup over? I understand that $\alpha, \beta$ are multi-indices, and I understand the notation $$D^\beta : = \frac{\partial^{ |\beta|}}{\partial x_1^{\beta_1} \cdots \partial x_n^{\beta_n}}$$, but I just don't quite understand why these functions are "rapidly decreasing". So, an explanation of the definition and notation would be appreciated.

Answer 1

I just don't quite understand why these functions are "rapidly decreasing".

The reason for the terminology is that, if $\phi\in\mathcal{S}(\mathbb R^n)$, then each $D^\alpha \phi$ tends to $0$ faster than $|x|^{-N}$ for all $N\geq 0$ as $|x|\to\infty$. (Introduction to the Theory of Distributions, page 93)

See the proof here.