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Concept
Schwartz space
Summary
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.
Schwartz space is named after French mathematician Laurent Schwartz.
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(R^n, C) is a subspace of the function space C^∞(R^n, C) of smooth functions from R^n into C.
If α 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 (xαDβ) f has a maximum in Rn by the extreme value theorem.
Because the Schwartz space is a vector space, any polynomial can by multiplied by a factor for a real constant, to give an element of the Schwartz space. In particular, there is an embedding of polynomials inside a Schwartz space.
From Leibniz's rule, it follows that S(R^n) is also closed under pointwise multiplication:
If f, g ∈ S(R^n) then the product fg ∈ S(R^n).
The Fourier transform is a linear isomorphism F:S(R^n) → S(R^n).
If f ∈ S(R) then f is uniformly continuous on R.
S(R^n) is a distinguished locally convex Fréchet Schwartz TVS over the complex numbers.
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