Painless Gabor expansions on homogeneous manifolds

We discuss the construction of Gabor systems on a manifold M. For this purpose, we define systems of functions indexed by a position and a frequency variable, with the purpose of expanding arbitrary square-integrable functions in this system. It turns out that such expansions can be obtained rather effortlessly. Moreover, if the manifold in question is the quotient of a Lie group G by a compact subgroup, the action of G allows to construct in a natural way smooth Gabor systems that are meaningfully indexed by elements of the cotangent bundle. The associated Gabor transform isometrically maps arbitrary L2-functions on M to smooth functions on the cotangent bundle of the manifold, and intertwines the natural actions of G on L2(M) and L2(T∗M).