  A comparative study of the numerical renormalization group and non-crossing
approximation results for the spectral functions of the $U=\infty$ Anderson
impurity model is carried out. The non-crossing approximation is the simplest
conserving approximation and has led to useful insights into strongly
correlated models of magnetic impurities. At low energies and temperatures the
method is known to be inaccurate for dynamical properties due to the appearance
of singularities in the physical Green's functions. The problems in developing
alternative reliable theories for dynamical properties have made it difficult
to quantify these inaccuracies. As a first step in obtaining a theory which is
valid also in the low energy regime, we identify the origin of the problems
within the NCA. We show, by comparison with close to exact NRG calculations for
the auxiliary and physical particle spectral functions, that the main source of
error in the NCA is in the lack of vertex corrections in the convolution
formulae for physical Green's functions. We show that the dynamics of the
auxiliary particles within NCA is essentially correct for a large parameter
region, including the physically interesting Kondo regime, for all energy
scales down to $T_{0}$, the low energy scale of the model, and often well below
this scale. Despite the satisfactory description of the auxiliary particle
dynamics, the physical spectral functions are not obtained accurately on scales
$\sim T_{0}$. Our results suggest that self--consistent conserving
approximations which include vertex terms may provide a highly accurate way of
dealing with strongly correlated systems at low temperatures.
