The Wiener-Lévy Theorem and Some of Its Converses

Abstract

Let G and Γ denote locally compact abelian groups, each the dual of the other. Let U ⊆ ℂ and let F be a complex-valued function defined on U. Suppose that $\hat{\mu}$ is a Fourier-Stieltjes transform on Γ with $\hat{\mu }$(Γ) ⊆ U. If F ∘ $\hat{\mu }$ is also a Fourier-Stieltjes transform, we say F operates on μ, and we let F º μ denote the measure whose transform is F ∘ $\hat{\mu }$. This chapter discusses necessary and sufficient conditions under which F operates on all μ that belong to varying classes of measures. These results have their origin in the theorem of Wiener and Lévy.