Schwartz space explained

In mathematics, Schwartz space

l{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 l{S} ^{*} of l{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

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

The Schwartz space or space of rapidly decreasing functions on

R ^{n} is the function space$$S\; \backslash left(\backslash mathbb^n,\; \backslash mathbb\backslash right)\; :=\; \backslash left\; \backslash ,$$where C ^{infty} ( R ^{n,} C ) is the function space of smooth functions from R ^{n} into C , and$$\backslash |f\backslash |\_:=\; \backslash sup\_\; \backslash left|\; \backslash boldsymbol^\backslash boldsymbol\; (\backslash boldsymbol^\backslash boldsymbol\; f)(\backslash boldsymbol)\; \backslash right|.$$ Here, \sup denotes the supremum, and we used multi-index notation, i.e. \boldsymbol{x} ^{\alpha _{1}} _{1} x ^{\alpha _{2}} _{2} \ldots x ^{\alpha _{n}} _{n} and D ^{\beta _{1}} _{1} \partial ^{\beta _{2}} _{2} \ldots \partial ^{\beta _{n}} _{n} .

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

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 l{S}( 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 Leibniz's rule, it follows that is also closed under pointwise multiplication:

If then the product .In particular, this implies that is an -algebra. More generally, if and is a bounded smooth function with bounded derivatives of all orders, then .

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

1. complete Hausdorff locally convex spaces,
2. nuclear Montel spaces,

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. Ultrabornological spaces,
2. reflexive barrelled Mackey spaces.

Relation of Schwartz spaces with other topological vector spaces

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

See also

• Bump function
• Schwartz–Bruhat function
• Nuclear space

References

Sources

• Book: Hörmander, L. . Lars Hörmander . The Analysis of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis) . 2nd . Berlin . Springer-Verlag . 1990 . 3-540-52343-X .
• Book: Reed, M. . B. . Simon . Methods of Modern Mathematical Physics: Functional Analysis I . Revised and enlarged . Academic Press . San Diego . 1980 . 0-12-585050-6 .
• Book: Stein, Elias M. . Rami . Shakarchi . Fourier Analysis: An Introduction (Princeton Lectures in Analysis I) . Princeton . Princeton University Press . 2003 . 0-691-11384-X .