Deforming the Window of a Gabor Frame: the Ellipsoid Method

TL;DR

This paper demonstrates how to modify the window of a Gabor frame using metaplectic operators, building on previous work on weak Hamiltonian deformations, with limited changes to the frame lattice.

Contribution

It introduces a method to deform Gabor frame windows via metaplectic operators while only altering finitely many lattice points.

Findings

Gabor frame windows can be deformed using metaplectic operators.

Finite modifications to the frame lattice are sufficient for deformation.

Builds on previous weak Hamiltonian deformation results.

Abstract

In a recent paper in Appl. Comput. Harmon. Anal. 38(2), 196--221 (2014) we have introduced and studied the notion of weak Hamiltonian deformation of a Gabor (=Weyl-Heisenberg) frame. In this Note we use these results to prove that one can modify the window of a Gabor frame using certain metaplectic operators provided that one modifies only a finite number of points of the frame lattice.