Schwartz space

In mathematics, Schwartz space is the function space of all functions whose derivatives of all orders 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 of , 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 be the set of non-negative integers, and for any , let be the -fold Cartesian product.

The Schwartz space or space of rapidly decreasing functions on is the function space where is the function space of smooth functions from into , and Here, denotes the supremum, and we used multi-index notation, i.e. and .

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 .

Examples of functions in the Schwartz space

• If is a multi-index, and a is a positive real number, then .
• Any smooth function with compact support is in . This is clear since any derivative of is continuous and supported in the support of , so has a maximum in by the extreme value theorem.
• Because the Schwartz space is a vector space, any polynomial can be multiplied by a factor for 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 is also closed under pointwise multiplication: implies . 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 Lipschitz continuous and hence 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,

1. ultrabornological spaces,
2. reflexive barrelled Mackey spaces.

Relation of Schwartz spaces with other topological vector spaces

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

See also

• Bump function
• Schwartz–Bruhat function
• Nuclear space

References

Sources