On the ideal structure of the algebra of radial functions

Schwartz, Alan

doi:10.1090/S0002-9939-1970-0265865-6

On the ideal structure of the algebra of radial functions
HTML articles powered by AMS MathViewer

by Alan Schwartz PDF

Proc. Amer. Math. Soc. 26 (1970), 621-624 Request permission

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

Abstract:

Let $L$ denote the convolution Banach algebra of integrable functions defined on ${R^n}$ and let ${L_r}$ consist of the subalgebra of radial functions. If $I$ is a closed ideal of $L$, the zero-set of $I$ is defined by $Z(I) = \{ y|\hat f(y) = 0{\text { for all }}f \in I\}$ where $\hat f$ is the Fourier transform of $f$. The following theorem is proved. If ${I_1}$ and ${I_2}$ are closed ideals of ${L_r}$ such that ${I_1} \subset {I_2}$ ($\subset$ denotes proper inclusion) then there is a closed ideal $I$ such that ${I_1} \subset I \subset {I_2}$.

References

S. Bochner and K. Chandrasekharan, Fourier Transforms, Annals of Mathematics Studies, No. 19, Princeton University Press, Princeton, N. J.; Oxford University Press, London, 1949. MR 0031582
Henry Helson, On the ideal structure of group algebras, Ark. Mat. 2 (1952), 83–86. MR 49912, DOI 10.1007/BF02591383
Elias M. Stein and Rami Shakarchi, Fourier analysis, Princeton Lectures in Analysis, vol. 1, Princeton University Press, Princeton, NJ, 2003. An introduction. MR 1970295
Sadahiro Saeki, An elementary proof of a theorem of Henry Helson, Tohoku Math. J. (2) 20 (1968), 244–247. MR 231139, DOI 10.2748/tmj/1178243181