Ignoring fine mathematical details in our explanation one can say that a system of the form is dual to if has an -convergent representation of the form

Any function generating a dual system is called a

Wexler and Raz have obtained an elegant formulation of the duality condition of the systems and . For the case of Gabor analysis over their basic result translates into the following characterization of duality with respect to given lattice constants

i.e., if and only if the two Weyl-Heisenberg systems generated from and respectively, with respect to the adjoint lattice constant , are biorthogonal to each other.

There are different ways to understand this *Wexler-Raz condition*
, one of them being a beautiful
representation of the Gabor frame operator due to Janssen,
who has shown that it can be written as a series of
time-frequency shifts along the adjoint lattice
with parameters .
This representation has also been derived at
about the same time by Daubechies, H. Landau and E. Landau
and by Ron and Shen. It should be noted that
the approaches used in these three references are quite different.

Ron and Shen have observed that there is an important duality between between oversampling and undersampling lattices (in a multi-dimensional setting). An -function generates a frame for given if and only if it generates a Riesz basis (for its closed linear span) for the lattice constants .

This survey would be incomplete without mentioning
a major tool in the analysis of Gabor systems, namely the
*Zak-transform*.
Actually, this transform (in the mathematical community known as the
*Weil-Brezin transform* [Fol89])
has been introduced by
Gelfand [Gel50] (1950), and was rediscovered
by A. Weil and independently by J. Zak
(Zak himself called it the -transform).
The Zak transform is in fact highly efficient for the case of
integer oversampling (i.e.,
), because in this case it
diagonalizes the frame operator.
In a generalized form it can also be used to study Gabor expansions for
rational oversampling. Recently other techniques (like the Kohn-Nirenberg
correspondence or,
for numerical purposes, the concept of unitary matrix
factorization) have turned out to be
powerful tools for the analysis of rationally oversampled Gabor frames.