  Traveling oscillating fronts (TOFs) are specific waves of the form $U_\star
(x,t) = e^{-i \omega t} V_\star(x - ct)$ with a profile $V_{\star}$ which
decays at $- \infty$ but approaches a nonzero limit at $+\infty$. TOFs usually
appear in complex Ginzburg Landau equations of the type $U_t = \alpha U_{xx} +
G(|U|^2)U$. In this paper we prove a theorem on the asymptotic stability of
TOFs, where we allow the initial perturbation to be the sum of an exponentially
localized part and a front-like part which approaches a small but nonzero limit
at $+ \infty$. The underlying assumptions guarantee that the operator, obtained
from linearizing about the TOF in a co-moving and co-rotating frame, has
essential spectrum touching the imaginary axis in a quadratic fashion and that
further isolated eigenvalues are bounded away from the imaginary axis. The
basic idea of the proof is to consider the problem in an extended phase space
which couples the wave dynamics on the real line to the ODE dynamics at $+
\infty$. Using slowly decaying exponential weights, the framework allows to
derive appropriate resolvent estimates, semigroup techniques, and Gronwall
estimates.
