space of rapidly decreasing functions


The function spaceMathworldPlanetmath of rapidly decreasing functions 𝒮 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 𝒮, that is, for tempered distributions.

Definition The space of rapidly decreasing functions on ℝn is the function space

𝒮(ℝn)={f∈C∞(ℝn)∣supx∈ℝn∣||f||α,β<∞for all multi-indicesα,β},

where C∞⁢(ℝn) is the set of smooth functionsMathworldPlanetmath from ℝn to ℂ, and

||f||α,β=||xα⁢Dβ⁢f||∞.

Here, ||⋅||∞ is the supremum norm, and we use multi-index notation. When the dimensionPlanetmathPlanetmathPlanetmath n is clear, it is convenient to write 𝒮=𝒮⁢(ℝn). The space 𝒮 is also called the Schwartz space, after Laurent Schwartz (1915-2002) [2].

0.0.1 Examples of functions in 𝒮

  1. 1.

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

    xi⁢exp⁡{-a⁢x2}∈𝒮.
  2. 2.

    Any smooth function with compact support f is in 𝒮. This is clear since any derivative of f is continuousMathworldPlanetmath, so xα⁢Dβ⁢f has a maximum in ℝn.

0.0.2 Properties

  1. 1.

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

  2. 2.

    Using Leibniz’ rule, it follows that 𝒮 is also closed under point-wise multiplication; if f,g∈𝒮, then f⁢g:x↦f⁢(x)⁢g⁢(x) is also in 𝒮.

  3. 3.

    For any 1≤p≤∞, we have [3]

    𝒮⊂Lp,

    and if p<∞, then 𝒮 is also dense in Lp.

  4. 4.

    The Fourier transform is a linear isomorphism 𝒮→𝒮.

References

  • 1 L. Hörmander, 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