Wavelet transform on the circle and the real line: A unified group-theoretical treatment

We present a unified group-theoretical derivation of the continuous wavelet transform (CWT) on the circle S-1 and the real line R, following the general formalism of coherent states (CS) associated to unitary square integrable (modulo a subgroup, possibly) representations of the group SL(2, R). A general procedure for obtaining unitary representations of a group G of affine transformations on a space of signals L-2(X, dx) is described, relating carrier spaces X to (first- or higher-order) "polarization subalgebras" P-X. We also provide explicit admissibility and continuous frame conditions for wavelets on S-1 and discuss the Euclidean limit in terms of group contraction. (C) 2006 Elsevier Inc. All rights reserved.