space of rapidly decreasing functions
The function space^{} of rapidly decreasing functions $\mathcal{S}$ has the important property that the Fourier transform^{} is an endomorphism on this space. This property enables one, by duality, to define the Fourier transform for elements in the dual space^{} of $\mathcal{S}$, that is, for tempered distributions.
Definition The space of rapidly decreasing functions on ${\mathbb{R}}^{n}$ is the function space
$$ 
where ${C}^{\mathrm{\infty}}({\mathbb{R}}^{n})$ is the set of smooth functions^{} from ${\mathbb{R}}^{n}$ to $\u2102$, and
$${f}_{\alpha ,\beta}={{x}^{\alpha}{D}^{\beta}f}_{\mathrm{\infty}}.$$ 
Here, $\cdot {}_{\mathrm{\infty}}$ is the supremum norm, and we use multiindex notation. When the dimension^{} $n$ is clear, it is convenient to write $\mathcal{S}=\mathcal{S}({\mathbb{R}}^{n})$. The space $\mathcal{S}$ is also called the Schwartz space, after Laurent Schwartz (19152002) [2].
0.0.1 Examples of functions in $\mathcal{S}$

1.
If $i$ is a multiindex, and $a$ is a positive real number, then
$${x}^{i}\mathrm{exp}\{a{x}^{2}\}\in \mathcal{S}.$$ 
2.
Any smooth function with compact support $f$ is in $\mathcal{S}$. This is clear since any derivative of $f$ is continuous^{}, so ${x}^{\alpha}{D}^{\beta}f$ has a maximum in ${\mathbb{R}}^{n}$.
0.0.2 Properties

1.
$\mathcal{S}$ is a complex vector space. In other words, $\mathcal{S}$ is closed under pointwise addition and under multiplication by a complex scalar.

2.
Using Leibniz’ rule, it follows that $\mathcal{S}$ is also closed under pointwise multiplication; if $f,g\in \mathcal{S}$, then $fg:x\mapsto f(x)g(x)$ is also in $\mathcal{S}$.

3.
For any $1\le p\le \mathrm{\infty}$, we have [3]
$$\mathcal{S}\subset {L}^{p},$$ and if $$, then $\mathcal{S}$ is also dense in ${L}^{p}$.

4.
The Fourier transform is a linear isomorphism $\mathcal{S}\to \mathcal{S}$.
References
 1 L. Hörmander, The Analysis of Linear Partial Differential Operators I, (Distribution^{} theory and Fourier Analysis), 2nd ed, SpringerVerlag, 1990.
 2 The MacTutor History of Mathematics archive, http://wwwgap.dcs.stand.ac.uk/ history/Mathematicians/Schwartz.htmlLaurent Schwartz
 3 M. Reed, B. Simon, Methods of Modern Mathematical Physics: Functional Analysis^{} I, Revised and enlarged edition, Academic Press, 1980.
 4 Wikipedia, http://en.wikipedia.org/wiki/Tempered_distributionTempered distributions
Title  space of rapidly decreasing functions 

Canonical name  SpaceOfRapidlyDecreasingFunctions 
Date of creation  20130322 13:44:50 
Last modified on  20130322 13:44:50 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  8 
Author  matte (1858) 
Entry type  Definition 
Classification  msc 46F05 
Synonym  Schwartz space 
Related topic  DiscreteTimeFourierTransformInRelationWithItsContinousTimeFourierTransfrom 