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

S. Dahlke, F. De Mari, E. De Vito, M. Hansen, M. Hasannasab, M. Quellmalz, G. Steidl, G. Teschke

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.

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

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.

Paper Structure

This paper contains 10 sections, 10 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Continuous wavelet frames on homogeneous spaces
  4. Group theoretic aspects
  5. Main result
  6. General necessary and sufficient admissibility condition
  7. Upper frame bound
  8. Stereographic projection and moment conditions
  9. Lower frame bound
  10. Alternative proof of Proposition \ref{['prop-strictly-positive']}

Key Result

Theorem 3.1

Fix $\eta\in L_2(\mathbb S^2)$ such that Then there is a constant $B>0$ such that for all $\phi\in L^2(\mathbb S^2)$ if and only if Under this condition, the frame operator $A_\eta: L_2(\mathbb S^2)\to L_2(\mathbb S^2)$ defined by is bounded, and it is boundedly invertible if and only if the function

Theorems & Definitions (23)

  • Remark 2.1
  • Theorem 3.1: Main Result
  • Lemma 3.2
  • Theorem 4.1
  • proof
  • Remark 4.2: Relation to AV99
  • Proposition 5.1
  • proof
  • Lemma 5.2
  • proof
  • ...and 13 more