  The phonon spectral function of the one-dimensional Holstein model is
obtained within weak and strong-coupling approximations based on analytical
self-energy calculations. The characteristic excitations found in the limit of
small charge-carrier density are related to the known (electronic) spectral
properties of Holstein polarons such as the polaron band dispersion. Particular
emphasis is laid on the different physics occurring in the adiabatic and
anti-adiabatic regimes, respectively. Comparison is made with a cluster
approach exploiting exact numerical results on small systems to yield an
approximation for the thermodynamic limit. This method, similar to cluster
perturbation theory, confirms the analytical findings, and yields accurate
results also in the intermediate-coupling regime.
