Continuous Wavelet Frames on the Sphere: The Group-Theoretic Approach Revisited

TL;DR

This paper revisits the group-theoretic construction of continuous wavelet frames on the sphere, providing necessary and sufficient conditions for functions to form such frames and strengthening previous results with detailed proofs.

Contribution

It offers a complete characterization of functions that generate wavelet frames on the sphere using group-theoretic methods, improving upon prior work.

Findings

Derived necessary and sufficient conditions for wavelet frames on the sphere.

Provided a complete, detailed proof of the characterization.

Strengthened previous theoretical results in the field.

Abstract

In \cite{AV99}, Antoine and Vandergheynst propose a group-theoretic approach to continuous wavelet frames on the sphere. The frame is constructed from a single so-called admissible function by applying the unitary operators associated to a representation of the Lorentz group, which is square-integrable modulo the nilpotent factor of the Iwasawa decomposition. We prove necessary and sufficient conditions for functions on the sphere, which ensure that the corresponding system is a frame. We strengthen a similar result in \cite{AV99} by providing a complete and detailed proof.