Tiling functions and Gabor orthonormal basis.

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. [ABSTRACT FROM AUTHOR]