Reproducing subgroups of Sp(2,R). Part II: admissible vectors

TL;DR

This paper classifies certain Lie subgroups of Sp(2,R) and identifies which allow the metaplectic representation to produce reproducing formulas, characterizing admissible vectors including wavelets and shearlets.

Contribution

It extends the classification of subgroups of Sp(2,R) and provides a detailed characterization of admissible vectors for the metaplectic representation.

Findings

Identified subgroups of Sp(2,R) with reproducing formulas

Characterized admissible vectors via a generalized Calderón equation

Included new examples like shearlets and directional wavelets

Abstract

In part I we introduced the class ${\mathcal E}_2$ of Lie subgroups of $Sp(2,\R)$ and obtained a classification up to conjugation (Theorem 1.1). Here, we determine for which of these groups the restriction of the metaplectic representation gives rise to a reproducing formula. In all the positive cases we characterize the admissible vectors with a generalized Calder\'on equation. They include products of 1D-wavelets, directional wavelets, shearlets, and many new examples.