  We investigate the structural properties of dual systems for nonstationary
Gabor frames. In particular, we prove that some inverse nonstationary Gabor
frame operators admit a Walnut-like representation, i.e. the operator acting on
a function can be described by weighted translates of that function, even when
the original frame operator is not diagonal. In this case, which only occurs
when compactly supported window functions are used, the canonical dual frame
partially inherits the structure of the original frame, with differences that
we describe in detail. Moreover, we determine a sufficient condition for a pair
of nonstationary Gabor frames to form dual frames. The equivalence of this
condition to the duality of the involved systems is shown under some weak
restrictions. It is then applied in a simple setup, to prove the existence of
dual pairs of nonstationary Gabor systems with non-diagonal frame operator.
  A discussion of the results, restricted to the classical case of regular
Gabor systems, precedes the statement of the general results. Here, we also
explore a connection to recent work of Christensen, Kim and Kim on Gabor frames
with compactly supported window function.
