  In long-baseline interferometry, bandwidth smearing of an extended source
occurs at finite bandwidth when its different components produce interference
packets that only partially overlap. In this case, traditional model fitting or
image reconstruction using standard formulas and tools lead to biased results.
  We propose and implement a method to overcome this effect by calculating
analytically a corrective term for the conventional interferometric
observables: the visibility amplitude and closure phase. For that purpose, we
model the interferogram taking into account the finite bandwidth and the
instrumental differential phase. We obtain generic expressions for the
visibility and closure phase in the case of temporally-modulated
interferograms, either processed using Fourier analysis or with the ABCD
method. The expressions can be used to fit arbitrary models to the data. We
then apply our results to the search and characterisation of stellar companions
with PIONIER at the Very Large Telescope Interferometer, assessing the bias on
observables and model-fitted parameters of a binary star. Finally, we consider
the role of the atmosphere, with an analytic model to identify the main
contributions to bias and also a numerical simulation of the turbulence.
  In addition to the analytic expressions, the main results of our study are:
the chromatic dispersion in the beam transport in the instrument has a strong
impact on the closure phase and introduces additional biases even at
separations where smearing is not expected to play an important role; the
atmospheric turbulence introduces additional biases when smearing is present,
but the impact is important only at very low spectral resolution; the bias on
the observables strongly depends on the recombination scheme and data
processing; the goodness of fits is improved by modelling a Gaussian bandpass
as long as the smearing is moderate.
