Extensions of the Heisenberg Group by Dilations and Frames

Extensions of the Heisenberg Group by Dilations and Frames

Two standard tools for signal analysis are the short{time Fourier transform and the continuous wavelet transform. These tools arise as matrix coeecients of square integrable representations of the Heisenberg and aane groups respectively, and discrete frame decompositions of L 2 arise from approximations of corresponding reproducing formulae. Here we study two groups, the so-called aane Weyl-Hei...

Stable transitivity of Heisenberg group extensions of hyperbolic systems

We consider skew-extensions with fiber the standard real Heisenberg group Hn of a uniformly hyperbolic dynamical system. We show that among the C extensions (r > 0) that avoid an obvious obstruction, those that are topologically transitive contain an open and dense set. More precisely, we show that an Hn-extension is transitive if and only if the R-extension given by the abelianization of Hn is...

Oversampling of wavelet frames for real dilations

We generalize the Second Oversampling Theorem for wavelet frames and dual wavelet frames from the setting of integer dilations to real dilations. We also study the relationship between dilation matrix oversampling of semi-orthogonal Parseval wavelet frames and the additional shift invariance gain of the core subspace.

Transitivity of Heisenberg group extensions of hyperbolic systems

We show that among Cr extensions (r > 0) of a uniformly hyperbolic dynamical system with fiber the standard real Heisenberg group Hn of dimension 2n + 1, those that avoid an obvious obstruction to topological transitivity are generically topological transitive. Moreover, if one considers extensions with fiber a connected nilpotent Lie group with compact commutator subgroup (for example, Hn/Z), ...

φ-factorable operators and Weyl-Heisenberg frames on LCA groups