Gabor frames for model sets

TL;DR

This paper extends key concepts of Gabor analysis from lattices to model sets, establishing foundational identities and relations, and explores the implications of dual Gabor systems on model set density.

Contribution

It generalizes fundamental Gabor analysis concepts to model sets and introduces a bracket product, connecting almost periodic functions and Poisson summation in this context.

Findings

Established Gabor analysis identities for model sets

Developed a bracket product linking almost periodic functions and Poisson summation

Proved density condition for dual Gabor systems in model sets

Abstract

We generalize three main concepts of Gabor analysis for lattices to the setting of model sets: Fundamental Identity of Gabor Analysis, Janssen's representation of the frame operator and Wexler-Raz biorthogonality relations. Utilizing the connection between model sets and almost periodic functions, as well as Poisson's summations formula for model sets we develop a form of a bracket product that plays a central role in our approach. Furthermore, we show that, if a Gabor system for a model set admits a dual which is of Gabor type, then the density of the model set has to be greater than one.