Inmathematics,Schwartz space${\displaystyle \mathcal{S}}$ is thefunction space of allfunctions whosederivatives of all orders arerapidly decreasing. This space has the important property that theFourier transform is anautomorphism on this space. This property enables one, by duality, to define the Fourier transform for elements in the dual space${\displaystyle {\mathcal{S}}^{\ast }}$ of⁠${\displaystyle \mathcal{S}}$⁠, that is, fortempered distributions. A function in the Schwartz space is sometimes called aSchwartz function.

Schwartz space is named after French mathematicianLaurent Schwartz.

Let${\displaystyle \mathbb{N}}$ be theset of non-negativeintegers, and for any⁠${\displaystyle n\in \mathbb{N}}$⁠, let${\displaystyle {\mathbb{N}}^n:=\underset{n\text{ times}}{\underbrace{\mathbb{N}\times \cdots \times \mathbb{N}}}}$ be the⁠${\displaystyle n}$⁠-foldCartesian product.

TheSchwartz space orspace of rapidly decreasing functions on${\displaystyle {\mathbb{R}}^n}$ is the function space where${\displaystyle C^{\mathrm{\infty }}({\mathbb{R}}^n,\mathbb{C})}$ is the function space ofsmooth functions from${\displaystyle {\mathbb{R}}^n}$ into⁠${\displaystyle \mathbb{C}}$⁠, and$${\displaystyle \Vert f{\Vert }_{\mathit{\alpha },\mathit{\beta }}:=\underset{\mathit{x}\in {\mathbb{R}}^n}{sup}\left|{\mathit{x}}^{\mathit{\alpha }}({\mathit{D}}^{\mathit{\beta }}f)(\mathit{x})\right|.}$$ Here,${\displaystyle sup}$ denotes thesupremum, and we usedmulti-index notation, i.e.${\displaystyle {\mathit{x}}^{\mathit{\alpha }}:=x_1^{{\alpha }_1}{x}_2^{{\alpha }_2}\dots x_n^{{\alpha }_n}}$ and⁠${\displaystyle D^{\mathit{\beta }}:={\mathrm{\partial }}_1^{{\beta }_1}{\mathrm{\partial }}_2^{{\beta }_2}\dots {\mathrm{\partial }}_n^{{\beta }_n}}$⁠.

To put common language to this definition, one could consider a rapidly decreasing function as essentially a function${\displaystyle f}$ such that⁠${\displaystyle f(x)}$⁠,⁠${\displaystyle f^{\prime }(x)}$⁠,⁠${\displaystyle f^{\mathrm{\prime }\mathrm{\prime }}(x)}$⁠, ... all exist everywhere on${\displaystyle \mathbb{R}}$ and go to zero as${\displaystyle x\to \pm \mathrm{\infty }}$ faster than any reciprocal power of⁠${\displaystyle x}$⁠. In particular,${\displaystyle \mathcal{S}\left({\mathbb{R}}^n,\mathbb{C}\right)}$ is asubspace of⁠${\displaystyle C^{\mathrm{\infty }}({\mathbb{R}}^n,\mathbb{C})}$⁠.

• If${\displaystyle \mathit{\alpha }}$ is a multi-index, anda is a positivereal number, then⁠${\displaystyle {\mathit{x}}^{\mathit{\alpha }}{e}^{-a|\mathit{x}{|}^2}\in \mathcal{S}({\mathbb{R}}^n)}$⁠.
• Any smooth function${\displaystyle f}$ withcompact support is in⁠${\displaystyle \mathcal{S}\left({\mathbb{R}}^n\right)}$⁠. This is clear since any derivative of${\displaystyle f}$ iscontinuous and supported in the support of⁠${\displaystyle f}$⁠, so${\displaystyle ({\mathit{x}}^{\mathit{\alpha }}{\mathit{D}}^{\mathit{\alpha }})f}$ has a maximum in${\displaystyle {\mathbb{R}}^n}$ by theextreme value theorem.
• Because the Schwartz space is a vector space, any polynomial${\displaystyle \varphi (\mathit{x})}$ can be multiplied by a factor${\displaystyle e^{-a|\mathit{x}{|}^2}}$ for${\displaystyle a>0}$ a real constant, to give an element of the Schwartz space. In particular, there is an embedding of polynomials into a Schwartz space.

• FromLeibniz's rule, it follows that${\displaystyle \mathcal{S}\left({\mathbb{R}}^n\right)}$ is also closed underpointwise multiplication:⁠${\displaystyle f,g\in \mathcal{S}\left({\mathbb{R}}^n\right)}$⁠ implies⁠${\displaystyle fg\in \mathcal{S}\left({\mathbb{R}}^n\right)}$⁠. In particular, this implies that${\displaystyle \mathcal{S}\left({\mathbb{R}}^n\right)}$ is an${\displaystyle \mathbb{R}}$-algebra. More generally, if${\displaystyle f\in \mathcal{S}\left(\mathbb{R}\right)}$ and${\displaystyle H}$ is a bounded smooth function with bounded derivatives of all orders, then⁠${\displaystyle fH\in \mathcal{S}\left(\mathbb{R}\right)}$⁠.
• The Fourier transform is alinear isomorphism⁠${\displaystyle \mathcal{F}:\mathcal{S}\left({\mathbb{R}}^n\right)\to \mathcal{S}\left({\mathbb{R}}^n\right)}$⁠.
• If${\displaystyle f\in \mathcal{S}\left({\mathbb{R}}^n\right)}$ then${\displaystyle f}$ isLipschitz continuous and henceuniformly continuous on⁠${\displaystyle {\mathbb{R}}^n}$⁠.
• ${\displaystyle \mathcal{S}\left({\mathbb{R}}^n\right)}$ is adistinguishedlocally convexFréchetSchwartz TVS over thecomplex numbers.
• Both${\displaystyle \mathcal{S}\left({\mathbb{R}}^n\right)}$and itsstrong dual space are also:

1. completeHausdorfflocally convex spaces,
2. nuclearMontel spaces,
3. ultrabornological spaces,
4. reflexivebarrelledMackey spaces.

• If⁠⁠, then${\displaystyle \mathcal{S}\left({\mathbb{R}}^n\right)}$ is adense subset of⁠${\displaystyle L^p({\mathbb{R}}^n)}$⁠.
• The space of allbump functions,⁠${\displaystyle C_{\text{c}}^{\mathrm{\infty }}\left({\mathbb{R}}^n\right)}$⁠, is included in⁠${\displaystyle \mathcal{S}\left({\mathbb{R}}^n\right)}$⁠.

• Bump function
• Schwartz–Bruhat function
• Nuclear space

• Hörmander, L. (1990).The Analysis of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis) (2nd ed.). Berlin: Springer-Verlag.ISBN 3-540-52343-X.
• Reed, M.; Simon, B. (1980).Methods of Modern Mathematical Physics: Functional Analysis I (Revised and enlarged ed.). San Diego: Academic Press.ISBN 0-12-585050-6.
• Stein, Elias M.; Shakarchi, Rami (2003).Fourier Analysis: An Introduction (Princeton Lectures in Analysis I). Princeton: Princeton University Press.ISBN 0-691-11384-X.
• Trèves, François (2006) [1967].Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications.ISBN 978-0-486-45352-1.OCLC 853623322.

This article incorporates material from Space of rapidly decreasing functions onPlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.

• Topological vector spaces
• Smooth functions
• Fourier analysis
• Function spaces
• Schwartz distributions

• Articles with short description
• Short description matches Wikidata
• Wikipedia articles incorporating text from PlanetMath