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Schwartz space

## Summary

In mathematics, Schwartz space ${\displaystyle {\mathcal {S}}}$ 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 ${\displaystyle {\mathcal {S}}^{*}}$ of ${\displaystyle {\mathcal {S}}}$, that is, for tempered distributions. A function in the Schwartz space is sometimes called a Schwartz function.

A two-dimensional Gaussian function is an example of a rapidly decreasing function.

Schwartz space is named after French mathematician Laurent Schwartz.

## Definition

### Motivation

The idea behind the Schwartz space is to consider the set of all smooth functions on ${\displaystyle \mathbb {R} ^{n}}$ which decrease rapidly. This is encoded by considering all possible derivatives ${\displaystyle D^{\alpha }f=\partial _{x_{1}}^{\alpha _{1}}\cdots \partial _{x_{n}}^{\alpha _{n}}f}$ (with multi-index ${\displaystyle \alpha }$) on a smooth complex-valued function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {C} }$ and the supremum of all possible values of ${\displaystyle D^{\alpha }f}$ multiplied by any monomial ${\displaystyle x^{\beta }}$ and bounding them. This restriction is encoded as the inequality

${\displaystyle \sup _{x\in \mathbb {R} ^{n}}|x^{\beta }D^{\alpha }f|<\infty }$
Notice if we only required the derivatives to be bounded, i.e.,
${\displaystyle \sup _{x\in \mathbb {R} ^{n}}|D^{\alpha }f|<\infty }$
this would imply all possible derivatives of a smooth function must be bounded by some constant ${\displaystyle c_{\alpha }}$, so
${\displaystyle \sup _{x\in \mathbb {R} ^{n}}|D^{\alpha }f|=c_{\alpha }<\infty }$
For example, the smooth complex-valued function ${\displaystyle f\in C^{\infty }(\mathbb {R} )}$ with ${\displaystyle f(x)=x^{2}}$ gives ${\displaystyle D^{1}f(x)=2x}$, which is an unbounded function, so any polynomial could not be in this space. But, if we require in addition the original inequality, then this result is even stronger because it implies the inequality
${\displaystyle D^{\alpha }f<{\frac {k_{\alpha ,\beta }}{x^{\beta }}}}$
for any ${\displaystyle \beta }$ and some constant ${\displaystyle k_{\alpha ,\beta }}$ since
${\displaystyle x^{\beta }\cdot D^{\alpha }f
This demonstrates the growth of all derivatives of ${\displaystyle f}$ must be far lesser than the inverse of any monomial.

### Definition

Let ${\displaystyle \mathbb {N} }$ be the set of non-negative integers, and for any ${\displaystyle n\in \mathbb {N} }$, let ${\displaystyle \mathbb {N} ^{n}:=\underbrace {\mathbb {N} \times \dots \times \mathbb {N} } _{n{\text{ times}}}}$ be the n-fold Cartesian product. The Schwartz space or space of rapidly decreasing functions on ${\displaystyle \mathbb {R} ^{n}}$ is the function space

${\displaystyle S\left(\mathbb {R} ^{n},\mathbb {C} \right):=\left\{f\in C^{\infty }(\mathbb {R} ^{n},\mathbb {C} )\mid \forall \alpha ,\beta \in \mathbb {N} ^{n},\|f\|_{\alpha ,\beta }<\infty \right\},}$
where ${\displaystyle C^{\infty }(\mathbb {R} ^{n},\mathbb {C} )}$ is the function space of smooth functions from ${\displaystyle \mathbb {R} ^{n}}$ into ${\displaystyle \mathbb {C} }$, and
${\displaystyle \|f\|_{\alpha ,\beta }:=\sup _{x\in \mathbb {R} ^{n}}\left|x^{\alpha }(D^{\beta }f)(x)\right|.}$
Here, ${\displaystyle \sup }$ denotes the supremum, and we use multi-index notation.

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→ ±∞ faster than any reciprocal power of x. In particular, S(Rn, C) is a subspace of the function space C(Rn, C) of smooth functions from Rn into C.

## Examples of functions in the Schwartz space

• If α is a multi-index, and a is a positive real number, then
${\displaystyle x^{\alpha }e^{-a|x|^{2}}\in {\mathcal {S}}(\mathbf {R} ^{n}).}$
• 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 (xαDβ) f has a maximum in Rn by the extreme value theorem.
• Because the Schwartz space is a vector space, any polynomial ${\displaystyle \phi (x^{\alpha })}$ can by multiplied by a factor ${\displaystyle e^{-ax^{2}}}$ for ${\displaystyle a>0}$ 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

It is known that in the dual space of any Montel space, a sequence converges in the strong dual topology if and only if it converges in the weak* topology,[1]

### Relation of Schwartz spaces with other topological vector spaces

• If 1 ≤ p ≤ ∞, then 𝒮(Rn) ⊂ Lp(Rn).
• If 1 ≤ p < ∞, then 𝒮(Rn) is dense in Lp(Rn).
• The space of all bump functions, C
c
(Rn)
, is included in 𝒮(Rn).