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Space Of Rapidly Decreasing Functions

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

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 $\mathcal{S}$

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

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 [3]
$\displaystyle \mathcal{S}\subset L^p,$
where $L^p(\mathbb{R}^n)$ is the space of -integrable functions on $\mathbb{R}^n$ . Functions in $\mathcal{S}$ are also bounded functions.
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

1	Types Of Homomorphisms
1	Scalar
1	Vector Space
1	Differntiable Function
1	Compact
1	Continuous
1	Complex
1	Real Number
1	Property
1	Obvious
1	Operation
1	Relation Theory
1	Multiplication
1	Addition
2	Increasingdecreasingmonotone Function
2	Derivative
2	Banach Space
3	Function Spaces
4	Polarity
4	Fourier Transform
4	Bounded Function
5	Support
5	Lp Space
5	Positive Element
6	Multi Index Notation
6	Distribution
7	Linear Isomorphism
10	Differentiation Under Integral Sign