  We propose and analyze a variational wave function for a
population-imbalanced one-dimensional Fermi gas that allows for
Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) type pairing correlations among the two
fermion species, while also accounting for the harmonic confining potential. In
the strongly interacting regime, we find large spatial oscillations of the
order parameter, indicative of an FFLO state. The obtained density profiles
versus imbalance are consistent with recent experimental results as well as
with theoretical calculations based on combining Bethe ansatz with the local
density approximation. Although we find no signature of the FFLO state in the
densities of the two fermion species, we show that the oscillations of the
order parameter appear in density-density correlations, both in-situ and after
free expansion. Furthermore, above a critical polarization, the value of which
depends on the interaction, we find the unpaired Fermi-gas state to be
energetically more favorable.
