Reproducing subgroups of $Sp(2,\mathbb{R})$. Part I: algebraic classification

TL;DR

This paper provides an algebraic classification of connected Lie subgroups of the symplectic group $Sp(2, eal)$ in block lower triangular form, aiding unified approaches in 2D signal analysis.

Contribution

It offers a comprehensive classification of certain subgroups of $Sp(2, eal)$, facilitating advances in continuous 2D signal analysis methods.

Findings

Classified subgroups up to conjugation within $Sp(2, eal)$

Identified subgroups relevant for wavelet and shearlet analysis

Established a foundation for unified 2D signal analysis approaches

Abstract

We classify the connected Lie subgroups of the symplectic group $Sp(2,\mathbb{R})$ whose elements are matrices in block lower triangular form. The classification is up to conjugation within $Sp(2,\mathbb{R})$. Their study is motivated by the need of a unified approach to continuous 2D signal analyses, as those provided by wavelets and shearlets.