Schwartz space

In mathematics, Schwartz space ${\mathcal {S}}$ is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. This property enables one, by duality, to define the Fourier transform for elements in the dual space ${\mathcal {S}}^{*}$ of ${\mathcal {S}}$ , that is, for

tempered distributions

. A function in the Schwartz space is sometimes called a Schwartz function.

Schwartz space is named after French mathematician Laurent Schwartz.

Definition

Let $\mathbb {N}$ be the set of non-negative integers, and for any $n\in \mathbb {N}$ , let $\mathbb {N} ^{n}:=\underbrace {\mathbb {N} \times \dots \times \mathbb {N} } _{n{\text{ times}}}$ be the n-fold Cartesian product.

The Schwartz space or space of rapidly decreasing functions on $\mathbb {R} ^{n}$ is the function space

S\left(\mathbb {R} ^{n},\mathbb {C} \right):=\left\{f\in C^{\infty }(\mathbb {R} ^{n},\mathbb {C} )\mid \forall {\boldsymbol {\alpha }},{\boldsymbol {\beta }}\in \mathbb {N} ^{n},\|f\|_{{\boldsymbol {\alpha }},{\boldsymbol {\beta }}}<\infty \right\},

where

C^{\infty }(\mathbb {R} ^{n},\mathbb {C} )

is the function space of

smooth functions

from

\mathbb {R} ^{n}

into

\mathbb {C}

, and

\|f\|_{{\boldsymbol {\alpha }},{\boldsymbol {\beta }}}:=\sup _{{\boldsymbol {x}}\in \mathbb {R} ^{n}}\left|{\boldsymbol {x}}^{\boldsymbol {\alpha }}({\boldsymbol {D}}^{\boldsymbol {\beta }}f)({\boldsymbol {x}})\right|.

Here,

\sup

denotes the

supremum, and we used multi-index notation

, i.e.

{\boldsymbol {x}}^{\boldsymbol {\alpha }}:=x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}}\ldots x_{n}^{\alpha _{n}}

and

D^{\boldsymbol {\beta }}:=\partial _{1}^{\beta _{1}}\partial _{2}^{\beta _{2}}\ldots \partial _{n}^{\beta _{n}}

.

To put common language to this definition, one could consider a rapidly decreasing function as essentially a function $f (x)$ such that $f (x)$ , $f'(x)$ , $f''(x)$ , ... all exist everywhere on $R$ and go to zero as $x$ $\to \pm\infty$ faster than any reciprocal power of $x$ . In particular, $S$ ( $R$ ^$n$, $C$ ) is a subspace of the function space $C$ ^$\infty$( $R$ ^$n$, $C$ ) of smooth functions from $R n$ into $C$ .

Examples of functions in the Schwartz space

If ${\boldsymbol {\alpha }}$ is a multi-index, and a is a positive real number, then
${\boldsymbol {x}}^{\boldsymbol {\alpha }}e^{-a|x|^{2}}\in {\mathcal {S}}(\mathbf {R} ^{n}).$
Any smooth function f with
compact support is in S(Rⁿ). This is clear since any derivative of f is continuous
and supported in the support of f, so ( ${\boldsymbol {x}}^{\boldsymbol {\alpha }}{\boldsymbol {D}}^{\boldsymbol {\alpha }})f$ has a maximum in Rⁿ by the extreme value theorem.
Because the Schwartz space is a vector space, any polynomial $\phi ({\boldsymbol {x}}^{\boldsymbol {\alpha }})$ can be multiplied by a factor $e^{-ax^{2}}$ for $a>0$ a real constant, to give an element of the Schwartz space. In particular, there is an embedding of polynomials inside a Schwartz space.

Properties

Analytic properties

From
pointwise multiplication
:
If $f, g \in 𝒮(R n)$ then the product $fg \in 𝒮(R n)$ .

In particular, this implies that $𝒮(R n)$ is an $R$ -algebra. More generally, if $f \in 𝒮(R)$ and $H$ is a bounded smooth function with bounded derivatives of all orders, then $fH \in 𝒮(R)$ .

The Fourier transform is a
linear isomorphism
$F:𝒮(R n) \to 𝒮(R n)$ .
If $f \in 𝒮(R)$ then $f$ is
uniformly continuous
on $R$ .
$𝒮(R n)$ is a distinguished locally convex Fréchet Schwartz TVS over the complex numbers.
Both $𝒮(R n)$ and its strong dual space are also: