Fourier Analysis on ℝn

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


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 .


Fourier Transform Smooth Function Sobolev Space Weak Type Schwartz Space 
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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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