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Fourier Analysis on ℝn

  • Michael Ruzhansky
  • Ville Turunen
Part of the Pseudo-Differential Operators book series (PDO, volume 2)

Abstract

In this chapter we review basic elements of Fourier analysis on ℝ n . Consequently, we introduce spaces of distributions, putting emphasis on the space of tempered distributions S′(ℝ n ). Finally, we discuss Sobolev spaces and approximation of functions and distributions by smooth functions. Throughout, we fix the measure on ℝ n to be Lebesgue measure. For convenience, we may repeat a few definitions in the context of ℝ n although they may have already appeared in Chapter C on measure theory. From this point of view, the present chapter can be read essentially independently. The notation used in this chapter and also in Chapter 2 is 〈ξ〉 = (1 + |ξ|2)1/2 where |ξ| = (ξ12 + ξ n 2)1/2, ξ ∈ ℝ n .

Keywords

Fourier Transform Smooth Function Sobolev Space Weak Type Schwartz Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag AG 2010

Authors and Affiliations

  • Michael Ruzhansky
    • 1
  • Ville Turunen
    • 2
  1. 1.Department of MathematicsImperial College LondonLondonUK
  2. 2.Institute of MathematicsHelsinki University of TechnologyFinland

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