In mathematics, 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 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

be the set of non-negative integers, and for any be the -fold Cartesian product.

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

such that , , faster than any reciprocal power of is a subspace of .

Examples of functions in the Schwartz space

• is a multi-index, and a is a positive real number, then .

• with compact support is in is continuous and supported in the support of by the extreme value theorem.

• 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 is also closed under pointwise multiplication: implies is a bounded smooth function with bounded derivatives of all orders, then .

• The Fourier transform is a linear isomorphism .

• is Lipschitz continuous and hence uniformly continuous on .

• is a distinguished locally convex Fréchet Schwartz TVS over the complex numbers.

• and its strong dual space are also:

• complete Hausdorff locally convex spaces,

• nuclear Montel spaces,

• ultrabornological spaces,

• reflexive barrelled Mackey spaces.

Relation of Schwartz spaces with other topological vector spaces

• If 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