  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.
