  We use the method of group contractions to relate wavelets analysis and Gabor
analysis. Wavelets analysis is associated with unitary irreducible
representations of the affine group while Gabor analysis is associated with
unitary irreducible representations of the Heisenberg group. We obtain unitary
irreducible representations of the Heisenberg group as contractions of
representations of the extended affine group. Furthermore, we use these
contractions to relate the two analyses, namely we contract coherent states,
resolutions of the identity, and tight frames. In order to obtain the standard
Gabor frame we construct a family of time localized wavelets frames that
contract to that Gabor frame. Starting from a standard wavelets frame we
construct a family of frequency localized wavelets frames that contract to a
nonstandard Gabor frame. In particular we deform Gabor frames to wavelets
frames.
