Gabor analysis as contraction of wavelets analysis

Abstract

We use the method of group contractions to relate wavelet analysis and Gabor analysis. Wavelet analysis is associated with unitary irreducible representations of the affine group while the 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 wavelet frames that contract to that Gabor frame. Starting from a standard wavelet frame, we construct a family of frequency localized wavelet frames that contract to a nonstandard Gabor frame. In particular, we deform Gabor frames to wavelet frames.