  We study the reduced Beurling spectra $sp_{\Cal {A},V} (F)$ of functions $F
\in L^1_{loc} (\jj,X)$ relative to certain function spaces $\Cal{A}\st
L^{\infty}(\jj,X)$ and $V\st L^1 (\r)$, where $\jj$ is $\r_+$ or $\r$ and $X$
is a Banach space. We show that if $F$ is bounded or slowly oscillating on
$\jj$ with $0\not\in sp_{\A,\f} (F)$, where $\A$ is $\{0\}$ or $C_0 (\jj,X)$
for example and $\f=\f(\r)$, then $F$ is ergodic. This result is new even for
$F\in BUC(\jj,X)$ and $\A= C_0(\jj,X)$. If $F$ is ergodic and belongs to the
space $ \f'_{ar}(\jj,X)$ of absolutely regular distributions and if
$sp_{C_0(\jj,X),\f} (F)=\emptyset$, then $\frak{F}*\psi \in C_0(\r,X)$ for all
$\psi\in \f(\r)$. Here, $\frak{F}|\jj =F$ and $\frak{F}|(\r\setminus\jj) =0$.
We show that tauberian theorems for Laplace transforms follow from results
about the reduced spectrum. Our results are more widely applicable than those
of previous authors. We demonstrate this and the sharpness of our results
through examples.

Functional Analysis