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space of rapidly decreasing functions (Definition)

The function space of rapidly decreasing functions $ \mathcal{S}$ 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.

Definition The space of rapidly decreasing functions on $ \mathbb{R}^n$ is the function space

$\displaystyle \mathcal{S}(\mathbb{R}^n)=\{ f \in C^\infty(\mathbb{R}^n) \mid \sup_{x\in \mathbb{R}^n} \mid \, \vert\vert f\vert\vert _{\alpha,\beta} < \infty\,$   for all multi-indices$\displaystyle \, \alpha, \beta \},$      

where $ C^\infty(\mathbb{R}^n)$ is the set of smooth functions from $ \mathbb{R}^n$ to $ \mathbb{C}$, and
$\displaystyle \vert\vert f\vert\vert _{\alpha,\beta}=\vert\vert x^\alpha D^\beta f\vert\vert _\infty.$
Here, $ \vert\vert\cdot\vert\vert _\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)$. The space $ \mathcal{S}$ is also called the Schwartz space, after Laurent Schwartz (1915-2002) [2].

Examples of functions in $ \mathcal{S}$

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

Properties

  1. $ \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.
  2. 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}$.
  3. For any $ 1\le p\le \infty$, we have [3]
    $\displaystyle \mathcal{S}\subset L^p,$
    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.
  4. The Fourier transform is a linear isomorphism $ \mathcal{S}\to\mathcal{S}$.

Bibliography

1
L. Hörmander, The Analysis of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis), 2nd ed, Springer-Verlag, 1990.
2
The MacTutor History of Mathematics archive, Laurent Schwartz
3
M. Reed, B. Simon, Methods of Modern Mathematical Physics: Functional Analysis I, Revised and enlarged edition, Academic Press, 1980.
4
Wikipedia, Tempered distributions



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See Also: discrete time Fourier transform in relation with continuous time Fourier transform

Other names:  Schwartz space
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Cross-references: linear isomorphism, bounded functions, Leibniz rule, scalar, multiplication, addition, closed under, vector space, complex, continuous, derivative, support, compact, real number, positive, multi-index, clear, multi-index notation, supremum norm, smooth functions, distributions, dual space, duality, endomorphism, Fourier transform, property, functions, decreasing, function space
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This is version 4 of space of rapidly decreasing functions, born on 2003-07-13, modified 2006-01-16.
Object id is 4444, canonical name is SpaceOfRapidlyDecreasingFunctions.
Accessed 5964 times total.

Classification:
AMS MSC: 46F05 (Functional analysis :: Distributions, generalized functions, distribution spaces :: Topological linear spaces of test functions, distributions and ultradistributions)
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