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Bochner–Minlos Theorem and Quaternion Fourier Transform

  • S. Georgiev
  • J. Morais
  • K. I. Kou
  • W. Sprößig
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

There have been several attempts in the literature to generalize the classical Fourier transform by making use of the Hamiltonian quaternion algebra. The first part of this chapter features certain properties of the asymptotic behaviour of the quaternion Fourier transform. In the second part we introduce the quaternion Fourier transform of a probability measure, and we establish some of its basic properties. In the final analysis, we introduce the notion of positive definite measure, and we set out to extend the classical Bochner–Minlos theorem to the framework of quaternion analysis.

Keywords

Quaternion analysis quaternion Fourier transform asymptotic behaviour positive definitely measure Bochner–Minlos theorem 

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© Springer Basel 2013

Authors and Affiliations

  • S. Georgiev
    • 1
  • J. Morais
    • 2
  • K. I. Kou
    • 3
  • W. Sprößig
    • 4
  1. 1.Department of Differential EquationsUniversity of SofiaSofiaBulgaria
  2. 2.Centro de Investigação e Desenvolvimento em Matemática e AplicaçÕes (CIDMA)Universidade de AveiroAveiroPortugal
  3. 3.Department of Mathematics, Faculty of Science and TechnologyUniversity of MacauMacauPeople’s Republic of China
  4. 4.Freiberg University of Mining and TechnologyFreibergGermany

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