  The solution of a multi-frequency 1d inverse medium problem consists of
recovering the refractive index of a medium from measurements of the scattered
waves for multiple frequencies. In this paper, rigorous stability estimates are
derived when the frequency takes value in a bounded interval.It is showed that
the ill-posedness of the inverse medium problem decreases as the width of the
frequency interval becomes larger. More precisely, under certain regularity
assumptions on the refractive index, the estimates indicate that the power in
H\"older stability is an increasing function of the largest value in the
frequency band. Finally, a Lipschitz stability estimate is obtained for the
observable part of the medium function defined through a truncated trace
formula.
