Schwartz space

In mathematics, Schwartz space is the function space of rapidly decreasing functions. This space has the important property that the Fourier transform is an endomorphism 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. Schwartz space is named in honour of Laurent Schwartz. A function in the Schwartz space is sometimes called a "Schwartz function".

Definition

The Schwartz space or space of rapidly decreasing functions $mathcal\{S\}$ on Rⁿ is the function space

:

where α, β are multi-indices, C^∞(Rⁿ) is the set of smooth functions from Rⁿ to C, and

:

Here, $||cdot||\_infty$ is the supremum norm, and we use multi-index notation. When the dimension "n" is clear, it is convenient to write $mathcal\{S\}=mathcal\{S\}(mathbb\{R\}^n)$.

Examples of functions in S

* If "i" is a multi-index, and "a" is a positive real number, then:$x^i\; e^\{-a\; x^2\}\; in\; mathcal\{S\}\; (mathbb\{R\}).$

* Any smooth function "f" with compact support is in $mathcal\{S\}$. This is clear since any derivative of "f" is continuous, so ("x"^α D^β) "f" has a maximum in Rⁿ.

Properties

* $mathcal\{S\}$ is a Fréchet space over the complex numbers.

* Using Leibniz' rule, it follows that $mathcal\{S\}$ is also closed under point-wise multiplication; if $f,g\; in\; mathcal\{S\}$, then $fg:\; xmapsto\; f(x)g(x)$ is also in $mathcal\{S\}$.

* For any 1 ≤ "p" ≤ ∞, we have $mathcal\{S\}subset\; L^p,$ where "L"^p(Rⁿ) is the space of "p"-integrable functions on Rⁿ. In particular, functions in $mathcal\{S\}$ are bounded (Reed & Simon 1980).

* The Fourier transform is a linear isomorphism $mathcal\{S\}\; o\; mathcal\{S\}$.

References

* L. Hörmander, "The Analysis of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis)", 2nd ed, Springer-Verlag, 1990.
* M. Reed, B. Simon, "Methods of Modern Mathematical Physics: Functional Analysis I, Revised and enlarged edition", Academic Press, 1980.