Analytic Features of Reproducing Groups for the Metaplectic Representation

Abstract

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.

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

“…As pointed out in previous work [2,3,8], many known continuous formulae (notably those associated to wavelets, shearlets and some of their variants) arise in this way, or are at least equivalent to them via natural intertwining operators such as the Fourier transform, perhaps combined with geometric (affine) transformations of phase space. But much more is true.…”

“…Symmetry and finite dimensional geometry often play a prominent rôle in the way in which these directions are generated or selected, and hence, with this notion of signal analysis, topological transformation groups and their representations provide a natural setup. In particular, the restriction of the metaplectic representation of Sp(d, R) to its Lie subgroups produces a wealth of useful reproducing formulae [2,3], all based on linear geometric actions either in the time or in the frequency domain, and is thus one of the most natural environments both for a unified approach and for the search of new strategies. In fact, the deep connections of the metaplectic representation with harmonic analysis in phase space is thoroughly investigated [10,13], and one of the keys to its understanding is the Wigner transform.…”

Reproducing Groups for the Metaplectic Representation

“…A complete classification of reproducing subgroups in the case d = 1 is given in [5] and many interesting facts have been proved in [13] in a somewhat different setting. Some new results in higher dimensions are in [4]. For the relevance of the extended metaplectic representation in the context of harmonic analysis in phase space or time-frequency analysis, the reader is referred to [8] and [9], and the references therein.…”

Continuous Inversion Formulas for Multi-Dimensional Stockwell Transforms

“…It should also be mentioned that transforms closely related to the wavelet transforms and the metaplectic representation abound and can be found in the monographs [1,7], and the works [3,14].…”

“…In a nutshell, we have modified through appropriate normalizations in the L 2 norm the Stockwell transform so that it becomes the Gabor transform in the preceding section. The results and examples in this section are known as special cases of reproducing formulas obtained from the metaplectic representation in Chapter 4 of [7] and in [3]. In Section 5 we give some examples of matrices that guarantee the resolution of the identity formula for the associated Stockwell transform.…”