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

DOI:
https://doi.org/10.1090/S0002-9939-1971-0412736-9

MathSciNet review:
0412736

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Abstract | References | Similar Articles | Additional Information

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.

**[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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1971-0412736-9

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