Reproducing Groups for the Metaplectic Representation

Harmonic and Applied Analysis

Vito

2015

The role of unitary group representations in applied mathematics is manifold and has been frequently pointed out and exploited. In this chapter, we first review the basic notions and constructs of Lie theory and then present the main features of some of the most useful unitary representations, such as the wavelet representation of the affine group, the Schrödinger representation of the Heisenberg group, and the metaplectic representation. The emphasis is on reproducing formulae. In the last section we discuss a promising class of unitary representations arising by restricting the metaplectic representation to triangular subgroups of the symplectic group. This class includes many known important examples, like the shearlet representation, and others that have not been looked at from the point of view of possible applications, like the so-called Schrödingerlets.

Section: The Metaplectic Representationmentioning

confidence: 99%

The Use of Representations in Applied Harmonic Analysis

Harmonic and Applied Analysis

Vito

2015

“…The latter method is used in the next section, while the former is applied in Section 5. We stress that Theorem 1 admits other useful applications [6,7].…”

Section: Definition 2 We Say That a Connected Lie Subgroupmentioning

confidence: 99%

“…The proof of the following theorem is analogous to the proof of Theorem 3 and its details are given in [7].…”

Section: The Intertwining Operator and The Equivalence For T Ds(2)mentioning

confidence: 99%

See 1 more Smart Citation

Analytic Features of Reproducing Groups for the Metaplectic Representation

Cordero

Nowak

et al. 2006

J Fourier Anal Appl

We introduce the notion of admissible subgroup H of G = H d Sp(d, R) relative to the (extended) metaplectic representation µ e via the Wigner distribution. Under mild additional assumptions, it is shown to be equivalent to the fact that the identity f = H f, µ e (h)φ µ e (h)φ dh holds (weakly) for all f ∈ L 2 (R d ). We use this equivalence to exhibit classes of admissible subgroups of Sp(2, R). We also establish some connections with wavelet theory, i.e., with curvelet and contourlet frames.

“…Further, one seeks conditions that single out the analyzing windows, those for which (1.1) holds and, consequently, are named reproducing. A complete classification of reproducing subgroups when d = 1 is given in [5] and various examples in higher dimension have been worked out in [3,4].…”

Section: Introductionmentioning

confidence: 99%

Dimensional upper bounds for admissible subgroups for the metaplectic representation

Cordero

Mathematische Nachrichten

Nowak

et al. 2010

Self Cite

We prove dimensional upper bounds for admissible Lie subgroupsThe notion of admissibility captures natural geometric phenomena of the phase space and it is a sufficient condition for a subgroup to be reproducing. It is expressed in terms of absolutely convergent integrals of Wigner distributions, translated by the affine action of the subgroup. We show that dim H ≤ d 2 + 2d, whereas if H ⊂ Sp(d, R), then dim H ≤ d 2 + 1. Both bounds are shown to be optimal.