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 Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{S}

on Rn is the function space

Failed to parse (Missing texvc executable; please see math/README to configure.): \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

Failed to parse (Missing texvc executable; please see math/README to configure.): ||f||_{\alpha,\beta}=||x^\alpha D^\beta f||_\infty\,.

Here, Failed to parse (Missing texvc executable; please see math/README to configure.): ||\cdot||_\infty

is the supremum norm, and we use multi-index notation. When the dimension n is clear, it is convenient to write Failed to parse (Missing texvc executable; please see math/README to configure.): \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

Failed to parse (Missing texvc executable; please see math/README to configure.): x^i e^{-a x^2} \in \mathcal{S} (\mathbb{R}).

• Any smooth function f with compact support is in Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{S}

. This is clear since any derivative of f is continuous, so (x^α D^β) f has a maximum in Rⁿ.

Properties

• Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{S}

 is a Fréchet space over complex numbers. In other words, Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{S}
is closed under point-wise addition and under multiplication by a complex scalar.

• Using Leibniz' rule, it follows that Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{S}

is also closed under point-wise multiplication; if Failed to parse (Missing texvc executable; please see math/README to configure.): f,g \in \mathcal{S}

, then Failed to parse (Missing texvc executable; please see math/README to configure.): fg: x\mapsto f(x)g(x)

is also in Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{S}

.

• For any 1 ≤ p ≤ ∞, we have Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{S}\subset L^p,

where Lp(Rn) is the space of p-integrable functions on Rn. Functions in Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{S}
are also bounded functions (Reed & Simon 1980).

• The Fourier transform is a linear isomorphism Failed to parse (Missing texvc executable; please see math/README to configure.): \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 GFDL.fr:Espace de Schwartz it:Spazio di Schwartz