  We study the asymptotic decay of the Friedel density oscillations induced by
an open boundary in a one-dimensional chain of lattice fermions with a
short-range two-particle interaction. From Tomonaga-Luttinger liquid theory it
is known that the decay follows a power law, with an interaction dependent
exponent, which, for repulsive interactions, is larger than the noninteracting
value $-1$. We first investigate if this behavior can be captured by many-body
perturbation theory for either the Green function or the self-energy in lowest
order in the two-particle interaction. The analytic results of the former show
a logarithmic divergence indicative of the power law. One might hope that the
resummation of higher order terms inherent to the Dyson equation then leads to
a power law in the perturbation theory for the self-energy. However, the
numerical results do not support this. Next we use density functional theory
within the local-density approximation and an exchange-correlation functional
derived from the exact Bethe ansatz solution of the translational invariant
model. While the numerical results are consistent with power-law scaling if
systems of $10^4$ or more lattice sites are considered, the extracted exponent
is very close to the noninteracting value even for sizeable interactions.
