Wavelet Transform on the Circle and the Real Line: A Unified Group-Theoretical Treatment

TL;DR

This paper unifies the derivation of the continuous wavelet transform on the circle and real line using group theory, providing a general framework and explicit conditions for wavelet admissibility.

Contribution

It introduces a unified group-theoretical approach to derive wavelet transforms on different spaces, extending the formalism of coherent states for $SL(2,\mathbb{R})$.

Findings

Provides explicit admissibility conditions for wavelets on $\mathbb{S}^1$

Establishes a general procedure for unitary representations of affine groups

Discusses the Euclidean limit via group contraction

Abstract

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.