  In the last decade it has become clear that one of the central themes within
Gabor analysis (with respect to general time-frequency lattices) is a duality
theory for Gabor frames, including the Wexler-Raz biorthogonality condition,
the Ron-Shen's duality principle or Janssen's representation of a Gabor frame
operator. All these results are closely connected with the so-called {\it
Fundamental Identity of Gabor Analysis}, which we derive from an application of
Poisson's summation formula for the symplectic Fourier transform. The new
aspect of this presentation is the description of range of the validity of this
Fundamental Identity of Gabor Analysis using Wiener amalgam spaces and
Feichtinger's algebra $S_0(\R^d)$. Our approach is inspired by Rieffel's use of
the Fundamental Identity of Gabor Analysis in the study of operator algebras
generated by time-frequency shifts along a lattice, which was later
independently rediscovered by Tolmieri/Orr, Janssen, and Daubechies et al., and
Feichtinger/Kozek at various levels of generality, in the context of Gabor
analysis.
