  We study one-dimensional spinless fermions at zero and finite temperature T
using the density matrix renormalization group. We consider nearest as well as
next-nearest neighbor interactions; the latter render the system inaccessible
by a Bethe ansatz treatment. Using an infinite-system alogrithm we demonstrate
the emergence of Luttinger liquid physics at low energies for a variety of
static correlation functions as well as for thermodynamic properties. The
characteristic power law suppression of the momentum distribution n(k) function
at T=0 can be directly observed over several orders of magnitude. At finite
temperature, we show that n(k) obeys a scaling relation. The Luttinger liquid
parameter and the renormalized Fermi velocity can be extracted from the density
response function, the specific heat, and/or the susceptibility without the
need to carry out any finite-size analysis. We illustrate that the energy scale
below which Luttinger liquid power laws manifest vanishes as the half-filled
system is driven into a gapped phase by large interactions.
