Reproducing subgroups of $$Sp(2,\mathbb R )$$ S p ( 2 , R ) . Part II: admissible vectors

Abstract

In part I we introduced the class \(\mathcal{E }_2\) of Lie subgroups of \(Sp(2,\mathbb R )\) and obtained a classification up to conjugation ( see Theorem 6 below). 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 Calderón equation. They include products of 1D-wavelets, directional wavelets, shearlets, and many new examples.