  We discuss the concepts of pseudo-dual frames and approximately dual frames,
and illuminate their relationship to classical frames. Approximately dual
frames are easier to construct than the classical dual frames, and might be
tailored to yield almost perfect reconstruction.
  For approximately dual frames constructed via perturbation theory, we provide
a bound on the deviation from perfect reconstruction. An alternative bound is
derived for the rich class of Gabor frames, by using the Walnut representation
of the frame operator to estimate the deviation from equality in the duality
conditions.
  As illustration of the results, we construct explicit approximate duals of
Gabor frames generated by the Gaussian; these approximate duals yield almost
perfect reconstruction. Amazingly, the method applies also to certain Gabor
frames that are far from being tight.
