On the ideal structure of the algebra of radial functions

Author:
Alan Schwartz

Journal:
Proc. Amer. Math. Soc. **26** (1970), 621-624

MSC:
Primary 42.56

DOI:
https://doi.org/10.1090/S0002-9939-1970-0265865-6

Correction:
Proc. Amer. Math. Soc. **39** (1973), 288-294.

MathSciNet review:
0265865

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Abstract | References | Similar Articles | Additional Information

Abstract: Let denote the convolution Banach algebra of integrable functions defined on and let consist of the subalgebra of radial functions. If is a closed ideal of , the zero-set of is defined by where is the Fourier transform of . The following theorem is proved. If and are closed ideals of such that ( denotes proper inclusion) then there is a closed ideal such that .

**[1]**S. Bochner and K. Chandrasekharan,*Fourier transforms*, Ann. of Math. Studies, no. 19, Princeton Univ. Press, Princeton, N. J., 1949. MR**11**, 173. MR**0031582 (11:173d)****[2]**H. Helson,*On the ideal structure of group algebras*, Ark. Mat.**2**(1952), 83-86. MR**14**, 246. MR**0049912 (14:246d)****[3]**E. M. Stein and G. Weiss,*An introduction to Fourier analysis in Euclidean spaces*, Princeton Univ. Press, Princeton, N. J. (to appear). MR**1970295 (2004a:42001)****[4]**Sadahiro Saeki,*An elementary proof of a theorem of Henry Helson*, Tôhoku Math. J. (2)**20**(1968), 244-247. MR**37**#6694. MR**0231139 (37:6694)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1970-0265865-6

Keywords:
Convolution algebra,
Fourier transform,
ideal structure,
radial functions,
zero-sets

Article copyright:
© Copyright 1970
American Mathematical Society