Density results for Gabor systems associated with periodic subsets of the real line

The well-known density theorem for one-dimensional Gabor systems of the form {e2πimbxg(x−na)}m,n∈Z, where g∈L2(R)g∈L2(R), states that a necessary and sufficient condition for the existence of such a system whose linear span is dense in L2(R)L2(R), or which forms a frame for L2(R)L2(R), is that the density condition ab≤1 is satisfied. The main goal of this paper is to study the analogous problem for Gabor systems for which the window function gg vanishes outside a periodic set S⊂RS⊂R which is aZ-shift invariant. We obtain measure-theoretic conditions that are necessary and sufficient for the existence of a window gg such that the linear span of the corresponding Gabor system is dense in L2(S)L2(S). Moreover, we show that if this density condition holds, there exists, in fact, a measurable set E⊂RE⊂R with the property that the Gabor system associated with the same parameters a,ba,b and the window g=χEg=χE, forms a tight frame for L2(S)L2(S).