Wavelets on the N-sphere and Related Manifolds

We present a purely group-theoretical derivation of the continuous wavelet transform (CWT) on the (n ? 1)-sphere S n?1 , based on the construction of general coherent states associated to square integrable group representations. The parameter space of the CWT, Y SO(n)R + , is embedded into the generalized Lorentz group SO o (n; 1) via the Iwasawa decomposition, so that X ' SO o (n; 1)=N, where N ' R n?1. Then the CWT on S n?1 is derived from a suitable unitary representation of SO o (n; 1) acting in the space L 2 (S n?1 ; dd) of nite energy signals on S n?1 , which turns out to be square integrable over X. We nd a necessary condition for the admissibility of a wavelet, in the form of a zero mean condition, which entails all the usual ltering properties of the CWT. Next the Euclidean limit of this CWT on S n?1 is obtained by redoing the construction on a sphere of radius R and performing a group contraction for R ! 1, from which one recovers the usual CWT on at Euclidean space. Finally, we discuss the extension of this construction to the two-sheeted hyperboloid H n?1 SO o (n ? 1; 1)=SO(n ? 1) and some other Riemannian symmetric spaces.