Schwartz space

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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.

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[edit] Definition

The Schwartz space or space of rapidly decreasing functions \mathcal{S} on \mathbb{R}^n 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, and C^\infty(\mathbb{R}^n) is the set of smooth functions from \mathbb{R}^n to \mathbb{C}, and

||f||_{\alpha,\beta}=||x^\alpha D^\beta f||_\infty\,.

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).

[edit] Examples of functions in S

  • If i is a multi-index, and a is a positive real number, then
x^i \exp\{-a x^2\} \in \mathcal{S}.
  • Any smooth function f with compact support is in \mathcal{S}. This is clear since any derivative of f is continuous, so xαDβf has a maximum in \mathbb{R}^n.

[edit] Properties

  • \mathcal{S} is a complex vector space. In other words, \mathcal{S} is closed under point-wise addition and under multiplication by a complex scalar.
  • 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\le p\le \infty, we have \mathcal{S}\subset L^p, where L^p(\mathbb{R}^n) is the space of p-integrable functions on \mathbb{R}^n. Functions in \mathcal{S} are also bounded functions (Reed & Simon 1980).
  • The Fourier transform is a linear isomorphism \mathcal{S} \to \mathcal{S}.

[edit] 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.

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