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 S1 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 L2(X,dx) is described, relating carrier spaces X to (first- or higher-order) “polarization subalgebras” PX. We also provide explicit admissibility and continuous frame conditions for wavelets on S1 and discuss the Euclidean limit in terms of group contraction.