# space of rapidly decreasing functions

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) .

## 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 

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

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

Title space of rapidly decreasing functions SpaceOfRapidlyDecreasingFunctions 2013-03-22 13:44:50 2013-03-22 13:44:50 matte (1858) matte (1858) 8 matte (1858) Definition msc 46F05 Schwartz space DiscreteTimeFourierTransformInRelationWithItsContinousTimeFourierTransfrom