Reproducing Subgroups of $Sp(2,\mathbb{R})$ . Part I: Algebraic Classification

Abstract

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

A classification of continuous wavelet transforms in dimension three

“…Further (partial) classification results exist for abelian irreducibly admissible groups in arbitrary dimensions [18], and for the class of generalized shearlet dilation groups [25] . In a similar spirit to our paper, the papers [1,2] classified a class of subgroups of the metaplectic group that allow an inversion formula under the metaplectic representation. We are not aware of further sources dealing with the systematical construction and classification of irreducibly admissible matrix groups or certain subclasses thereof.…”

“…Using the structure of the Lie algebra, with straightforward calculations, we check that the span of the set {w, π(X)w, π(X) 2 w} is invariant under the action of π(h). Appealing to the irreducibility of π the elements w, π(X)w, 1 2 π(X) 2 w form a basis for the vector space V. With respect to this basis, we obtain Let h be the linear span of the matrices above. We shall show that this algebra is isomorphic to so(2, 1) ⊕ R where Indeed, with straightforward computations, we verify that…”

Continuous and discrete frames generated by the evolution flow of the Schrödinger equation

“…; ii) the "restriction" of the mock-metaplectic representation π to the fiber Φ −1 (1) and to the stability subgroup SO(d) is precisely ρ. Hence, (3.7) provides the decomposition of ρ into its irreducibles, all of them with multiplicity 1; iii) up to the normalization factor 1/ √ 2, the operator SF −1 coincides with the operator introduced in [3], whose main feature is that it decomposes π into its irreducibles, each of which is the canonical representation obtained by inducing the irreducible representation of R × SO(d) acting on H i as…”

Coorbit Spaces with Voice in a Fréchet Space

“…The main feature of this example is that once the admissibility conditions are worked out, it is relatively easy to exhibit kernels in Ş pPI L p pGq but hard to find a kernel in L 1 pGq. This example has shown up in the classification of reproducing triangular subgroups of Spp2, Rq, which was recently achieved in [38,18].…”