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space of rapidly decreasing functions


The function spaceMathworldPlanetmath of rapidly decreasing functions ๐’ฎ has the important property that the Fourier transformDlmfMathworldPlanetmath is an endomorphism on this space. This property enables one, by duality, to define the Fourier transform for elements in the dual spaceMathworldPlanetmathPlanetmath of ๐’ฎ, that is, for tempered distributions.

Definition The space of rapidly decreasing functions on โ„n is the function space

๐’ฎ(โ„n)={fโˆˆCโˆž(โ„n)โˆฃsupxโˆˆโ„nโˆฃ||f||ฮฑ,ฮฒ<โˆžfor all multi-indicesฮฑ,ฮฒ},

where Cโˆž(โ„n) is the set of smooth functionsMathworldPlanetmath from โ„n to โ„‚, and

||f||ฮฑ,ฮฒ=||xฮฑDฮฒf||โˆž.

Here, ||โ‹…||โˆž is the supremum norm, and we use multi-index notation. When the dimensionPlanetmathPlanetmathPlanetmath n is clear, it is convenient to write ๐’ฎ=๐’ฎ(โ„n). The space ๐’ฎ is also called the Schwartz space, after Laurent Schwartz (1915-2002) [2].

0.0.1 Examples of functions in ๐’ฎ

  1. 1.

    If i is a multi-index, and a is a positive real number, then

    xiexp{-ax2}โˆˆ๐’ฎ.
  2. 2.

    Any smooth function with compact support f is in ๐’ฎ. This is clear since any derivative of f is continuousMathworldPlanetmath, so xฮฑDฮฒf has a maximum in โ„n.

0.0.2 Properties

  1. 1.

    ๐’ฎ is a complex vector space. In other words, ๐’ฎ is closed under point-wise addition and under multiplication by a complex scalar.

  2. 2.

    Using Leibnizโ€™ rule, it follows that ๐’ฎ is also closed under point-wise multiplication; if f,gโˆˆ๐’ฎ, then fg:xโ†ฆf(x)g(x) is also in ๐’ฎ.

  3. 3.

    For any 1โ‰คpโ‰คโˆž, we have [3]

    ๐’ฎโŠ‚Lp,

    and if p<โˆž, then ๐’ฎ is also dense in Lp.

  4. 4.

    The Fourier transform is a linear isomorphism ๐’ฎโ†’๐’ฎ.

References

  • 1 L. Hรถrmander, The Analysis of Linear Partial Differential Operators I, (DistributionDlmfPlanetmath 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 AnalysisMathworldPlanetmath 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