Phase Space Decompositions: Local Fourier Analysis on Spheres

Continuous wavelet analysis and Gabor analysis have proven to be very useful tools for the analysis of signals in which local frequencies can be extracted. Both techniques can be described in the same footing using the theory of square-integrable group representations or derived theories. It is shown here how the same kind of techniques can be developed to construct phase-space representation theormes for functions deened on homogeneous spaces. As examples, the cases of the circle and the spheres are carried in details, and some explicit expressions are given in the one and two-dimensional cases.