# Tiling functions and Gabor orthonormal basis

## TL;DR

This paper investigates conditions under which Gabor orthonormal bases with specific characteristic function windows exist, linking their existence to tiling properties of the underlying set, and fully characterizing their structure.

## Contribution

It establishes a precise equivalence between the existence of such Gabor bases and the tiling of the line by the set W, providing a complete structural description.

## Key findings

- Gabor orthonormal bases exist if and only if W tiles the line.
- Complete characterization of the time-frequency shifts for these bases.
- Results apply to sets with a specific union of intervals of measure 1.

## Abstract

We study the existence of Gabor orthonormal bases with window the characteristic function of the set W=[0,a] U [b+a, b+1] of measure 1, with a, b>0. By the symmetries of the problem, we can restrict our attention to the case a<=1/2. We prove that either if a<1/2 or (a=1/2 and b>= 1/2) there exist such Gabor orthonormal bases, with window the characteristic function of the set W, if and only if W tiles the line. Furthermore, in both cases, we completely describe the structure of the set of time-frequency shifts associated to these bases