Frame set for Gabor systems with Haar window

TL;DR

This paper characterizes the full structure of the frame set for Gabor systems with Haar window using a novel piecewise linear transformation related to the three gap theorem, providing a complete and self-contained analysis of Gabor frame conditions.

Contribution

It introduces a new transformation-based approach to fully describe Gabor frame sets with Haar windows, improving classical criteria and offering a self-contained framework.

Findings

Complete characterization of the frame set for Haar Gabor systems.

A necessary and sufficient condition for Gabor systems to be frames.

Connection of the transformation to the three gap theorem.

Abstract

We show the full structure of the frame set for the Gabor system $\mathcal{G}(g;\alpha,\beta):=\{e^{-2\pi i m\beta\cdot}g(\cdot-n\alpha):m,n\in\Bbb Z\}$ with the window being the Haar function $g=-\chi_{[-1/2,0)}+\chi_{[0,1/2)}$. The strategy of this paper is to introduce the piecewise linear transformation $\mathcal{M}$ on the unit circle, and to provide a complete characterization of structures for its (symmetric) maximal invariant sets. This transformation is related to the famous three gap theorem of Steinhaus which may be of independent interest. Furthermore, a classical criterion on Gabor frames is improved, which allows us to establish {a} necessary and sufficient condition for the Gabor system $\mathcal{G}(g;\alpha,\beta)$ to be a frame, i.e., the symmetric invariant set of the transformation $\mathcal{M}$ is empty. Compared with the previous studies, the present paper provides a self-contained environment to study Gabor frames by a new perspective, which includes that the techniques developed here are new and all the proofs could be understood thoroughly by the readers without reference to the known results in the previous literature.