The Schwartz space $\mathcal{S}$ is the space of smooth, infinitely differentiable and rapidly decreasing functions on $\mathbb{R^n.}$ Using a set builder we can rewrite $\mathcal{S}$ as: $$ \mathcal{S}(\mathbb{R^n})=\{f\in C^{\infty}(\mathbb{R^n}):||f||_{\alpha,\beta}<\infty \,, \forall \alpha, \beta\} \tag{1} $$ Where $\alpha,\beta$ are multi-indices and $$ ||f||_{\alpha,\beta}=\sup_{x\in \mathbb{R^n}} \left|x^\alpha D^\beta f(x)\right| \tag{2} $$
Questions:
- How can one interpret intuitively the definition of the norm of $f$ given in $(2)?$ And how does it imply that such $f$'s (and all its derivatives) should be decaying?
- By "rapidly decreasing functions", is it hinted that the decay rate of such functions is exponential? (otherwise, "rapid" with respect to what?)
- Finally, how can one explain that any smooth function $f$ with compact support is in $\mathcal{S}(\mathbb{R^n})?$
From (2) it follows that $$ |D^\beta f(x)|\le\frac{\|f\|_{\alpha,\beta}}{|x|^\alpha}. $$ $f$ and all its derivatives decay at $\infty$ faster than any negative power.
The decay may not be exponential. Consider for instance $e^{-(1+x^2)^a}$ with $0<a<1/2$.
If the support of $f$ is $K$, then for all multi-indices $\alpha,\beta$ $$ \sup_{x\in K}\bigl|x^\alpha D^\beta f(x)\bigr|<\infty. $$