Schwartz space: Information from Answers.com

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.

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

1 Definition
2 Examples of functions in S
3 Properties
4 References

Definition

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

$\mathcal{S} \left(\mathbb{R}^n\right) = \{ f \in C^\infty(\mathbb{R}^n) \mid \|f\|_{\alpha,\beta} < \infty\, \forall \, \alpha, \beta \},$

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

$\|f\|_{\alpha,\beta}=\sup_{x\in\mathbf{R}^n}|x^\alpha D^\beta f(x)|.$

Here, sup denotes the supremum, and we use multi-index notation. When the dimension n is clear, it is convenient to write $\mathcal{S}=\mathcal{S}(\mathbb{R}^n)$ . A rapidly decreasing function is in effect a function that goes to zero as $|x|\to\infty$ faster than any inverse power of x, as do all its derivatives.

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 $fg: x\mapsto f(x)g(x)$ 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 $\mathcal{S} \to \mathcal{S}$ .

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.

This article incorporates material from Space of rapidly decreasing functions on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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Schwartz space

Contents

Definition

Examples of functions in S

Properties

References