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 Sn-1. based on the construction of general coherent states associated to square integrable group representations. The parameter space of the CWT, X similar to SO(n)xR*(+), is embedded into the generalized Lorentz group SO0(n,1) via the Iwasawa decomposition, so that X similar or equal to SO0(n,1)IN, where N similar or equal to Rn-1. Then the CWT on Sn-1 is derived from a suitable unitary representation of SO0(n,1) acting in the space L-2(Sn-1,d mu) of finite energy signals on Sn-1, which turns out to be square integrable over X. We find a necessary condition for the admissibility of a wavelet, in the form of a zero mean condition, which entails all the usual filtering properties of the CWT. Next the Euclidean limit of this CWT on Sn-1 is obtained by redoing the construction on a sphere of radius R and performing a group contraction for R-->infinity, from which one recovers the usual CWT on flat Euclidean space. Finally, we discuss the extension of this construction to the two-sheeted hyperboloid Hn-1SO0(n-1,1)/SO(n-1) and some other Riemannian symmetric spaces. (C) 1998 American Institute of Physics. [S0022-2488(98)00308-9].