  For a Fell bundle $\mathcal{B}=\left\{B_{s}\right\}_{s \in G}$ over a
discrete group $G$, we use representations theory of $\mathcal{B}$ to construct
the Fourier and Fourier-Stieltjes spaces $A(\mathcal{B})$ and $B(\mathcal{B})$
of $\mathcal B$. When $\mathcal B$ is saturated we show $B(\mathcal{B})$ is
canonically isomorphic to the dual space of the cross sectional $C^{*}$-algebra
$C^{*}(\mathcal{B})$ of $\mathcal{B}$. When there is a compatible family of
co-multiplications on the fibers we show that $B(\mathcal{B})$ and
$A(\mathcal{B})$ are Banach algebras. This holds in particular if either the
fiber $B_e$ at identity is a Hopf $C^*$-algebra or $\mathcal{B}$ is the Fell
bundle of a $C^*$-dynamical system. When $A(\mathcal{B})$ is a Banach algebra
with bounded approximate identity, we show that $B(\mathcal{B})$ is the
multiplier algebra of $A(\mathcal{B})$. We prove a Leptin type theorem by
showing that amenability of $G$ implies the existence of bounded approximate
identity for $A(\mathcal{B})$ for bundles coming from a $C^*$-dynamical system
$(A,G,\gamma)$. The converse is left as an open problem.
