Gabor Frame Sets of Invariance - A Hamiltonian Approach to Gabor Frame Deformations

TL;DR

This paper introduces a Hamiltonian-based approach to identify families of Gabor frames with invariant bounds, demonstrating the existence of extensive sets of window functions and lattices that preserve frame properties.

Contribution

It constructs uncountable lattice families and countable Gaussian families that maintain Gabor frame bounds, revealing new invariance properties in Gabor analysis.

Findings

Uncountable families of lattices preserve frame bounds with a fixed Gaussian.

Countable families of Gaussians preserve frame bounds across different lattices.

The approach links Hamiltonian dynamics to Gabor frame invariance.

Abstract

In this work we study families of pairs of window functions and lattices which lead to Gabor frames which all possess the same frame bounds. To be more precise, for every generalized Gaussian $g$, we will construct an uncountable family of lattices $\lbrace \Lambda_\tau \rbrace$ such that each pairing of $g$ with some $\Lambda_\tau$ yields a Gabor frame, and all pairings yield the same frame bounds. On the other hand, for each lattice we will find a countable family of generalized Gaussians $\lbrace g_i \rbrace$ such that each pairing leaves the frame bounds invariant. Therefore, we are tempted to speak about "Gabor Frame Sets of Invariance".