Connection between Quantum tomography based on projective representation and wavelet transform in Banach spaces

The intimate connection between the Banach space wavelet reconstruction method based on the projective representation with both singular and nonsingular vacuum vectors, and some of well known quantum tomographies, such as: Dihedral group tomography, Discrete phase space tomography, tomography of SL(2,Z) subgroups, permutation group tomography, Homodyne tomography and SU(1,1) tomography is revealed. Also both the atomic decomposition and banach frame nature of these quantum tomographic examples is explained in details.