space of rapidly decreasing functions
The function space of rapidly decreasing functions ๐ฎ 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 ๐ฎ, that is, for tempered distributions.
Definition The space of rapidly decreasing functions on โn is the function space
๐ฎ(โn)={fโCโ(โn)โฃsupxโโnโฃ||f||ฮฑ,ฮฒ<โfor all multi-indicesฮฑ,ฮฒ}, |
where Cโ(โn) is the set of smooth functions from โn to โ, and
||f||ฮฑ,ฮฒ=||xฮฑDฮฒf||โ. |
Here, ||โ ||โ is the supremum norm, and we use multi-index notation. When the dimension n is clear, it is convenient to write ๐ฎ=๐ฎ(โn). The space ๐ฎ is also called the Schwartz space, after Laurent Schwartz (1915-2002) [2].
0.0.1 Examples of functions in ๐ฎ
-
1.
If i is a multi-index, and a is a positive real number, then
xiexp{-ax2}โ๐ฎ. -
2.
Any smooth function with compact support f is in ๐ฎ. This is clear since any derivative of f is continuous, so xฮฑDฮฒf has a maximum in โn.
0.0.2 Properties
-
1.
๐ฎ is a complex vector space. In other words, ๐ฎ is closed under point-wise addition and under multiplication by a complex scalar.
-
2.
Using Leibnizโ rule, it follows that ๐ฎ is also closed under point-wise multiplication; if f,gโ๐ฎ, then fg:xโฆf(x)g(x) is also in ๐ฎ.
- 3.
-
4.
The Fourier transform is a linear isomorphism ๐ฎโ๐ฎ.
References
- 1 L. Hรถrmander, 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, http://www-gap.dcs.st-and.ac.uk/ history/Mathematicians/Schwartz.htmlLaurent Schwartz
- 3 M. Reed, B. Simon, Methods of Modern Mathematical Physics: Functional Analysis I, Revised and enlarged edition, Academic Press, 1980.
- 4 Wikipedia, http://en.wikipedia.org/wiki/Tempered_distributionTempered distributions
Title | space of rapidly decreasing functions |
---|---|
Canonical name | SpaceOfRapidlyDecreasingFunctions |
Date of creation | 2013-03-22 13:44:50 |
Last modified on | 2013-03-22 13:44:50 |
Owner | matte (1858) |
Last modified by | matte (1858) |
Numerical id | 8 |
Author | matte (1858) |
Entry type | Definition |
Classification | msc 46F05 |
Synonym | Schwartz space |
Related topic | DiscreteTimeFourierTransformInRelationWithItsContinousTimeFourierTransfrom |