Frames of translates for model sets

TL;DR

This paper investigates the properties of translated functions along model sets, characterizing frame generators, and applying findings to Gabor systems, using almost periodic functions and Poisson summation techniques.

Contribution

It introduces a new approach to analyze irregular translates using the bracket product and establishes a density result for semi-regular Gabor frames.

Findings

Characterization of tight and dual frame generators for model set translates

Introduction of the bracket product for irregular translates

Density statement for semi-regular Gabor frames

Abstract

We study spanning properties of a family of functions translated along simple model sets. We characterize tight frame and dual frame generators for such irregular translates and we apply the results to Gabor systems. We use the connection between model sets and almost periodic functions and rely strongly on a Poisson summations formula for model sets to introduce the so-called bracket product, which then plays a crucial role in our approach. As a corollary to our main results we obtain a density statement for semi-regular Gabor frames.