space of rapidly decreasing functions
The function space of rapidly decreasing functions 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 , that is, for tempered distributions.
Definition The space of rapidly decreasing functions on is the function space
where is the set of smooth functions from to , and
Here, is the supremum norm, and we use multi-index notation. When the dimension is clear, it is convenient to write . The space is also called the Schwartz space, after Laurent Schwartz (1915-2002) [2].
0.0.1 Examples of functions in
-
1.
If is a multi-index, and is a positive real number, then
-
2.
Any smooth function with compact support is in . This is clear since any derivative of is continuous, so has a maximum in .
0.0.2 Properties
-
1.
is a complex vector space. In other words, is closed under point-wise addition and under multiplication by a complex scalar.
-
2.
Using Leibniz’ rule, it follows that is also closed under point-wise multiplication; if , then is also in .
- 3.
-
4.
The Fourier transform is a linear isomorphism .
References
- 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, 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 |