Functions that operate in the Fourier algebra of a compact group

Abstract

0. Only real-analytic functions operate in the Fourier algebra of any compact group that has an infinite abelian subgroup. This extends the theorems of Helson, Kahane, Katznelson, and Rudin [4] which apply to the algebra of absolutely convergent Fourier series on compact abelian groups. The Fourier algebra of a locally compact group has been studied by H. Mirkil [6], W. F. Stinespring [9], R. A. Mayer [5], C. Herz [3], and most thoroughly by P. Eymard [1]. We will state here the relevant definitions and facts, and prove that the restriction of the Fourier algebra to a closed subgroup is the Fourier algebra of the subgroup, and use this to lift up the theorem on operating functions.