Tiling functions and Gabor orthonormal basis

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