Schwartz space

In mathematics, Schwartz space

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

, 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.

be the set of non-negative integers, and for any

be the n-fold Cartesian product.

The Schwartz space or space of rapidly decreasing functions on

is the function space

α

β

α

β

is the function space of smooth functions from

α

β

α

β

denotes the supremum, and we used multi-index notation, i.e.

α

To put common language to this definition, one could consider a rapidly decreasing function as essentially a function f(x) such that f(x), f ′(x), f ′′(x), ... all exist everywhere on R and go to zero as x→ ±∞ faster than any reciprocal power of x.

In particular, 𝒮(Rn, C) is a subspace of the function space C∞(Rn, C) of smooth functions from Rn into C. In particular, this implies that 𝒮(Rn) is an R-algebra.

More generally, if f ∈ 𝒮(R) and H is a bounded smooth function with bounded derivatives of all orders, then fH ∈ 𝒮(R).

This article incorporates material from Space of rapidly decreasing functions on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.