Beurling Spectra of Functions on Locally Compact Abelian Groups

Abstract

Let $G$ be a locally compact abelian topological group. For locally bounded measurable functions $\phi :G\to \mathbb{C}$ we discuss notions of spectra for $\phi$ relative to subalgebras of $L^1(G)$. In particular we study polynomials on $G$ and determine their spectra. We also characterize the primary ideals of certain Beurling algebras $L_w^1(\mathbb{Z})$ on the group of integers $\mathbb{Z}$. This allows us to classify those elements of $L_w^1(G)$ that have finite spectrum. If $\phi$ is a uniformly continuous function whose differences are bounded, there is a Beurling algebra naturally associated with $\phi$. We give a condition on the spectrum of $\phi$ relative to this algebra which ensures that $\phi$ is bounded. Finally we give spectral conditions on a bounded function on $\mathbb{R}$ that ensure that its indefinite integral is bounded.

17 pages. Corrections and compliment to Monash Analysis Paper 88 (1993)