Fourier Transforms (II)

  • Toru Maruyama
Part of the Monographs in Mathematical Economics book series (MOME, volume 2)


In the previous chapter, we observed a peculiar relation between the smoothness and the rapidity of vanishing at infinity of a function f, as well as its Fourier transform \(\hat {f}\). Based upon this observation, we introduce an important function space \(\mathfrak {S}\), which is invariant under the Fourier transforms. We then proceed to \(\mathfrak {L}^2\)-theory of Fourier transforms due to M. Plancherel. As a simple application of Plancherel’s theory, we discuss how to solve integral equations of convolution type. Finally, a tempered distribution is defined as an element of \(\mathfrak {S}'\), and its Fourier transform is examined in detail.


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Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Toru Maruyama
    • 1
  1. 1.Professor EmeritusKeio UniversityTokyoJapan

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