ISSN:
1089-7658
Source:
AIP Digital Archive
Topics:
Mathematics
,
Physics
Notes:
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∼SO(n)×R*+, is embedded into the generalized Lorentz group SO0(n,1) via the Iwasawa decomposition, so that X(similar, equals)SO0(n,1)/N, where N(similar, equals)Rn−1. Then the CWT on Sn−1 is derived from a suitable unitary representation of SO0(n,1) acting in the space L2(Sn−1,dμ) 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→∞, 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−1∼SO0(n−1,1)/SO(n−1) and some other Riemannian symmetric spaces. © 1998 American Institute of Physics.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1063/1.532481
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