In mathematics, Schwartz space \mathcal{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 of \mathcal{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 \mathbb{N} be the set of non-negative integers, and for any, let be the n-fold Cartesian product. The Schwartz space or space of rapidly decreasing functions on is the function spacewhere is the function space of smooth functions from into \mathbb{C}, and Here, \sup 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 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( R n, C ) is a subspace of the function space C∞( R n, C ) of smooth functions from Rn into C .

Examples of functions in the Schwartz space

Properties

Analytic properties

𝒮(Rn) is also closed under pointwise multiplication: f, g ∈ 𝒮(Rn) then the product fg ∈ 𝒮(Rn) . 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) . F:𝒮(Rn) → 𝒮(Rn) . f ∈ 𝒮(R) then f is uniformly continuous on R . 𝒮(Rn) is a distinguished locally convex Fréchet Schwartz TVS over the complex numbers. 𝒮(Rn) and its strong dual space are also:

Relation of Schwartz spaces with other topological vector spaces

1 ≤ p ≤ ∞ , then 𝒮(Rn) ⊂ Lp(Rn) . 1 ≤ p < ∞ , then 𝒮(Rn) is dense in Lp(Rn) . C∞ c(Rn) , is included in 𝒮(Rn) .

Sources