# 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

 $\displaystyle\mathcal{S}(\mathbb{R}^{n})=\{f\in C^{\infty}(\mathbb{R}^{n})\mid% \sup_{x\in\mathbb{R}^{n}}\mid\,||f||_{\alpha,\beta}<\infty\,\mbox{for all % multi-indices}\,\alpha,\beta\},$

where $C^{\infty}(\mathbb{R}^{n})$ is the set of smooth functions from $\mathbb{R}^{n}$ to $\mathbb{C}$, and

 $||f||_{\alpha,\beta}=||x^{\alpha}D^{\beta}f||_{\infty}.$

Here, $||\cdot||_{\infty}$ is the supremum norm, and we use multi-index 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 (1915-2002) [2].

## 0.0.1 Examples of functions in $\mathcal{S}$

1. 1.

If $i$ is a multi-index, and $a$ is a positive real number, then

 $x^{i}\exp\{-ax^{2}\}\in\mathcal{S}.$
2. 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. 1.

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

2. 2.

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

3. 3.

For any $1\leq p\leq\infty$, we have [3]

 $\mathcal{S}\subset L^{p},$

and if $p<\infty$, then $\mathcal{S}$ is also dense in $L^{p}$.

4. 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, Springer-Verlag, 1990.
• 2 The MacTutor History of Mathematics archive, http://www-gap.dcs.st-and.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
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