Functions which operate in the Fourier algebra of a compact group

Author:
Daniel Rider

Journal:
Proc. Amer. Math. Soc. **28** (1971), 525-530

MSC:
Primary 46.80; Secondary 42.00

DOI:
https://doi.org/10.1090/S0002-9939-1971-0276792-3

MathSciNet review:
0276792

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be the Fourier algebra of a compact group . It is shown that a function defined on a closed convex subset of the plane operates in if and only if it is real analytic. This was shown by Helson, Kahane, Katznelson and Rudin when is locally compact and abelian and by Dunkl when is compact and contains an infinite abelian subgroup. A direct proof is given of the following lemma which is all that is needed in order to apply the proof of Helson, Kahane, Katznelson and Rudin ( is the Fourier algebra norm).

Lemma. *Let and be the set of such that is real and . Then*

**[1]**Charles W. Curtis and Irving Reiner,*Representation theory of finite groups and associative algebras*, Pure and Applied Mathematics, Vol. XI, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1962. MR**0144979****[2]**Charles F. Dunkl,*Functions that operate in the Fourier algebra of a compact group*, Proc. Amer. Math. Soc.**21**(1969), 540–544. MR**0239360**, https://doi.org/10.1090/S0002-9939-1969-0239360-6**[3]**Charles F. Dunkl and Donald E. Ramirez,*Topics in harmonic analysis*, Appleton-Century-Crofts [Meredith Corporation], New York, 1971. Appleton-Century Mathematics Series. MR**0454515****[4]**Pierre Eymard,*L’algèbre de Fourier d’un groupe localement compact*, Bull. Soc. Math. France**92**(1964), 181–236 (French). MR**0228628****[5]**Henry Helson, Jean-Pierre Kahane, Yitzhak Katznelson, and Walter Rudin,*The functions which operate on Fourier transforms*, Acta Math.**102**(1959), 135–157. MR**0116185**, https://doi.org/10.1007/BF02559571**[6]**B. Huppert,*Endliche Gruppen. I*, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR**0224703****[7]**Walter Rudin,*Fourier analysis on groups*, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley and Sons), New York-London, 1962. MR**0152834**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
46.80,
42.00

Retrieve articles in all journals with MSC: 46.80, 42.00

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1971-0276792-3

Keywords:
Fourier algebra,
compact group,
functions which operate,
real analytic

Article copyright:
© Copyright 1971
American Mathematical Society