  We study theoretically the dimensional crossover from a three-dimensional
elongated condensate to a one-dimensional condensate as the transverse degrees
of freedom get frozen by tight confinement, in the limit of small density
fluctuations, i.e. for a strongly degenerate gas. We compute analytically the
radially integrated density profile at low temperatures using a local density
approximation, and study the behavior of phase fluctuations with the transverse
confinement. Previous studies of phase fluctuations in trapped gases have
either focused on the 3D elongated regimes or on the 1D regime. The present
approach recovers these previous results and is able to interpolate between
them. We show in particular that in this strongly degenerate limit the shape of
the spatial correlation function is insensitive to the transverse regime of
confinement, pointing out to an almost universal behavior of phase fluctuations
in elongated traps.
