On new families of wavelets and Gabor analysis

We construct two new families of wavelets: One family of frames which is well suited for frequency localized signals and interpolates between the standard wavelet frames and a version of a Gabor type frame. The second family is well suited for time localized signals and interpolated between a version of a wavelet frame and a standard Gabor frame. In particular we approximate Gabor analysis by wavelets. Our construction is based on certain realizations of the unitary representations of the Heisenberg group and of the affine group on L^2(R). The main technical tool that we use for the interpolation procedures is contraction of Lie groups representations.