Geometric Aspects of Frame Representations of Abelian Groups

We consider frames arising from the action of a unitary representation of a discrete countable abelian group. We show that the range of the analysis operator can be determined by computing which characters appear in the representation. This allows one to compare the ranges of two such frames, which is useful for determining similarity and also for multiplexing schemes. Our results then partially extend to Bessel sequences arising from the action of the group. We apply the results to sampling on bandlimited functions and to wavelet and Weyl-Heisenberg frames. This yields a sufficient condition for two sampling transforms to have orthogonal ranges, and two analysis operators for wavelet and Weyl-Heisenberg frames to have orthogonal ranges. The sufficient condition is easy to compute in terms of the periodization of the Fourier transform of the frame generators.Comment: 20 pages; contact author: Eric Webe

Plancherel transform criteria for Weyl-Heisenberg frames with integer oversampling

We investigate the relevance of admissibility criteria based on Plancherel measure for the characterization of tight Weyl-Heisenberg frames with integer oversampling. For this purpose we observe that functions giving rise to such Weyl-Heisenberg frames are admissible with respect to the action of suitably defined type-I discrete group G. This allows to relate the construction of Weyl-Heisenberg frames to the Plancherel measure of G, which provides an alternative proof and a new interpretation of the well-known Zak transform based criterion for tight Weyl-Heisenberg frames with integer oversampling.Comment: 13 page

Symmetry, Hamiltonian Problems and Wavelets in Accelerator Physics

In this paper we consider applications of methods from wavelet analysis to nonlinear dynamical problems related to accelerator physics. In our approach we take into account underlying algebraical, geometrical and topological structures of corresponding problems.Comment: LaTeX2e, aipproc.sty, 25 pages, typos correcte

Nonstationary frames of translates and frames for the Weyl--Heisenberg group and the extended affine group

In this work, we analyze Gabor frames for the Weyl--Heisenberg group and wavelet frames for the extended affine group. Firstly, we give necessary and sufficient conditions for the existence of nonstationary frames of translates. Using these conditions, we give the existence of Gabor frames for the Weyl--Heisenberg group and wavelet frames for the extended affine group. We present a representation of functions in the closure of the linear span of a Gabor frame sequence in terms of the Fourier transform of window functions. Afterward, we give sufficient conditions with explicit frame bounds for a finite linear combination of Gabor frames to be a frame. It is illustrated that the frame bounds associated with finite linear combinations of frames can decrease the width of the frame, which increases the speed of convergence in the frame algorithm. We show that the canonical dual of frames of translates has the same structure. An approximation of inverse of the frame operator of nonstationary frames of translates is presented. It is shown that a nonstationary frame of translates is a Riesz basis if and only if it is linearly independent and satisfies approximation of the inverse frame operator. Finally, we give equivalent conditions for a nonstationary sequence of translates to be linearly independent