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 $\mathbb S^1$ and the real line $\mathbb{R}$, following the general formalism of Coherent States (CS) associated to unitary square integrable (modulo a subgroup, possibly) representations of the group $SL(2,\mathbb{R})$. 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'' ${\cal P}_X$. We also provide explicit admissibility and continuous frame conditions for wavelets on $\mathbb S^1$ and discuss the Euclidean limit in terms of group contraction.