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The Fourier transform is onto only when the group is finite

Author: Stephen H. Friedberg
Journal: Proc. Amer. Math. Soc. 27 (1971), 421-422
MSC: Primary 43A25
MathSciNet review: 0412736
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Abstract: A well-known result of Henry Helson is used to prove that a locally compact abelian group is finite if and only if the Fourier transform is a surjective map.

References [Enhancements On Off] (What's this?)

  • [1] R. Doss, Elementary proof of a theorem of Helson, Proc. Amer. Math. Soc. 27 (1971), 418-420. MR 0271644 (42:6527)
  • [2] M. Rajagopalan, Fourier transform in locally compact groups, Acta Sci. Math. (Szeged) 25 (1964), 86-89. MR 29 #6250. MR 0168995 (29:6250)
  • [3] W. Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Appl. Math., no. 12, Interscience, New York, 1962. MR 27 #2808. MR 0152834 (27:2808)
  • [4] I. E. Segal, The class of functions which are absolutely convergent Fourier transforms, Acta Sci. Math. (Szeged) 12 (1950), Pars B, 157-161. MR 12, 188; p. 1002. MR 0036943 (12:188d)

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Keywords: Locally compact abelian group, dual group, Fourier transform, extremally disconnected, Helson set, Haar measure, Tietze Extension Theorem
Article copyright: © Copyright 1971 American Mathematical Society

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