Schwartz space

Template:For multi In mathematics, Schwartz space $𝒮$ 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 ${𝒮}^{*}$ of $𝒮$, 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

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

The Schwartz space or space of rapidly decreasing functions on ${ℝ}^n$ is the function space$$𝒮({ℝ}^n,ℂ):=\{f\in C^{\mathrm{\infty }}({ℝ}^n,ℂ)\mid \mathrm{\forall }\mathit{\alpha },\mathit{\beta }\in {\mathbb{N}}^n,\Vert f{\Vert }_{\mathit{\alpha },\mathit{\beta }}<\mathrm{\infty }\},$$where $C^{\mathrm{\infty }}({ℝ}^n,ℂ)$ is the function space of smooth functions from ${ℝ}^n$ into $ℂ$, and$$\Vert f{\Vert }_{\mathit{\alpha },\mathit{\beta }}:=\underset{𝒙\in {ℝ}^n}{\sup }|{𝒙}^{\mathit{\alpha }}({𝑫}^{\mathit{\beta }}f)(𝒙)|.$$ Here, $\sup$ denotes the supremum, and we used multi-index notation, i.e. ${𝒙}^{\mathit{\alpha }}:=x_1^{{\alpha }_1}{x}_2^{{\alpha }_2}\dots x_n^{{\alpha }_n}$ and $D^{\mathit{\beta }}:=\underset{1}{\overset{{\beta }_1}{\partial }}\underset{2}{\overset{{\beta }_2}{\partial }}\dots \underset{n}{\overset{{\beta }_n}{\partial }}$.

To put common language to this definition, one could consider a rapidly decreasing function as essentially a function Template:Math such that Template:Math, Template:Math, Template:Math, ... all exist everywhere on Template:Math and go to zero as Template:MvarTemplate:Math faster than any reciprocal power of Template:Mvar. In particular, Template:Math is a subspace of the function space Template:MvarTemplate:Sup(Template:MathTemplate:Sup, Template:Math) of smooth functions from Template:Math into Template:Math.

Examples of functions in the Schwartz space

• If $\mathit{\alpha }$ is a multi-index, and a is a positive real number, then

  ${𝒙}^{\mathit{\alpha }}{e}^{-a|𝒙{|}^2}\in 𝒮({ℝ}^n).$

• Any smooth function f with compact support is in Template:Math. This is clear since any derivative of f is continuous and supported in the support of f, so (${𝒙}^{\mathit{\alpha }}{𝑫}^{\mathit{\alpha }})f$ has a maximum in Rⁿ by the extreme value theorem.
• Because the Schwartz space is a vector space, any polynomial $\varphi (𝒙)$ can be multiplied by a factor $e^{-a|𝒙{|}^2}$ for $a>0$ a real constant, to give an element of the Schwartz space. In particular, there is an embedding of polynomials into a Schwartz space.

Properties

Analytic properties

• From Leibniz's rule, it follows that Template:Math is also closed under pointwise multiplication:

  If Template:Math then the product Template:Math.

In particular, this implies that Template:Math is an Template:Math-algebra. More generally, if Template:Math and Template:Mvar is a bounded smooth function with bounded derivatives of all orders, then Template:Math.

• The Fourier transform is a linear isomorphism Template:Math.
• If Template:Math then Template:Mvar is Lipschitz continuous and hence uniformly continuous on Template:Math.
• Template:Math is a distinguished locally convex Fréchet Schwartz TVS over the complex numbers.
• Both Template:Math and its strong dual space are also:

1. complete Hausdorff locally convex spaces,
2. nuclear Montel spaces,
3. ultrabornological spaces,
4. reflexive barrelled Mackey spaces.

Relation of Schwartz spaces with other topological vector spaces

• If Template:Math, then Template:Math.
• If Template:Math, then Template:Math is dense in Template:Math.
• The space of all bump functions, Template:Math, is included in Template:Math.

See also

• Bump function
• Schwartz–Bruhat function
• Nuclear space

References

Sources

• Template:Cite book
• Template:Cite book
• Template:Cite book
• Template:Trèves François Topological vector spaces, distributions and kernels