Frames With Several Generators Associated with Weyl-Heisenberg Group and Extended Affine Group

TL;DR

This paper explores the construction of Gabor and wavelet frames for Weyl-Heisenberg and extended affine groups using group contraction methods, demonstrating their stability under perturbations.

Contribution

It introduces new methods to construct multi-generator Gabor and wavelet frames via group contraction, linking the two groups' frame structures.

Findings

Construction of Gabor frames with multiple generators for Weyl-Heisenberg group

Wavelet frames with multiple generators for extended affine group

Frames are stable under small perturbations

Abstract

We study the construction of Gabor frames and wavelet frames for Weyl-Heisenberg group and extended affine group by using contraction between the affine group and the Weyl-Heisenberg group due to Subag, Baruch, Birman and Mann. Firstly, we give construction of Gabor frames with several generators from a unitary irreducible representation associated to the Weyl-Heisenberg group. Wavelet frames with several generators associated with an extended affine group have been obtained. A relation between frames for Weyl-Heisenberg group and extended affine group is also discussed. Finally, we show that frames of Gabor and wavelet structure are stable under small perturbations.