Tiling functions and Gabor orthonormal basis

Abstract

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