Beurling Spectra of Functions on Locally Compact Abelian Groups

Abstract

Let $G$ be a locally compact abelian topological group. For locally bounded measurable functions $\varphi: G\to\Bbb {C}$ we discuss notions of spectra for $\varphi$ 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 $L_{w}^{1}(\Bbb Z)$ on the group of integers $\Bbb Z$. This allows us to classify those elements of $L_{w}^{1}(G)$ that have finite spectrum. If $\varphi$ is a uniformly continuous function whose differences are bounded, there is a Beurling algebra naturally associated with $\varphi$. We give a condition on the spectrum of $\varphi$ relative to this algebra which ensures that $\varphi$ is bounded. Finally we give spectral conditions on a bounded function on $\Bbb R$ that ensure that its indefinite integral is bounded.