Schwartz space

Function space of all functions whose derivatives are rapidly decreasing From Wikipedia, the free encyclopedia

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 of , that is, for tempered distributions. A function in the Schwartz space is sometimes called a Schwartz function.

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A two-dimensional Gaussian function is an example of a rapidly decreasing function.

Schwartz space is named after French mathematician Laurent Schwartz.

Definition

Summarize
Perspective

Let 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 , 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 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.

Examples of functions in the Schwartz space

  • If is a multi-index, and a is a positive real number, then
  • 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 ( has a maximum in Rn 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

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).

  1. complete Hausdorff locally convex spaces,
  2. nuclear Montel spaces,
  3. ultrabornological spaces,
  4. reflexive barrelled Mackey spaces.

Relation of Schwartz spaces with other topological vector spaces

  • If 1 ≤ p ≤ ∞, then 𝒮(Rn) ⊂ Lp(Rn).
  • If 1 ≤ p < ∞, then 𝒮(Rn) is dense in Lp(Rn).
  • The space of all bump functions, C
    c
    (Rn)
    , is included in 𝒮(Rn).

See also

References

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