  We study one-dimensional sideband cooling of Cesium atoms strongly confined
in a far-detuned optical lattice. The Lamb-Dicke regime is achieved in the
lattice direction whereas the transverse confinement is much weaker. The
employed sideband cooling method, first studied by Vuletic et al.\cite{Vule98},
uses Raman transitions between Zeeman levels and produces a spin-polarized
sample. We present a detailed study of this cooling method and investigate the
role of elastic collisions in the system. We accumulate $83(5)%$ of the atoms
in the vibrational ground state of the strongly confined motion, and elastic
collisions cool the transverse motion to a temperature of $2.8 \mu $K=$0.7
\hbar\omega_{\rm osc}/k_{\rm B}$, where $\omega_{\rm osc}$ is the oscillation
frequency in the strongly confined direction. The sample then approaches the
regime of a quasi-2D cold gas. We analyze the limits of this cooling method and
propose a dynamical change of the trapping potential as a mean of cooling the
atomic sample to still lower temperatures. Measurements of the rate of
thermalization between the weakly and strongly confined degrees of freedom are
compatible with the zero energy scattering resonance observed previously in
weak 3D traps. For the explored temperature range the measurements agree with
recent calculations of quasi-2D collisions\cite{Petr01}. Transparent analytical
models reproduce the expected behavior for $k_{\rm B}T \gg \hbar \omega_{\rm
osc}$ and also for $k_{\rm B}T \ll \hbar \omega_{\rm osc}$ where the 2D
features are prominent.
