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 S, that is, for tempered distributions. A function in the Schwartz space is sometimes called a Schwartz function.
is an example of a rapidly decreasing 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 n-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 use multi-index notation.
To put common language to this definition, one could consider a rapidly decreasing function as essentially a function f such that f, f ′, f′′,... all exist everywhere on R and go to zero as x → ±∞ faster than any inverse power of x. In particular, S is a subspace of the function space C∞ of smooth functions from Rn to C.Examples of functions in the Schwartz space
- If α is a multi-index, and a is a positive real number, then
- Any smooth function f with compact support is in S. This is clear since any derivative of f is continuous and supported in the support of f, so f has a maximum in Rn by the extreme value theorem.
Properties
- S is a Fréchet space over the complex numbers.
- From Leibniz' rule, it follows that S is also closed under pointwise multiplication: if f, g ∈ S then fg ∈ S.
- If 1 ≤ p ≤ ∞, then S ⊂ Lp.
- If 1 ≤ p < ∞, then S is dense in Lp.
- The space of all bump functions, C, is included in S.
- The Fourier transform is a linear isomorphism S → S.
- If f ∈ S, then f is uniformly continuous on R.
