Density of Gabor Schauder bases

A Gabor system is a fixed set of time-frequency shifts G(g, Lambda) = {e(2 pi ib.x)g(x - a)}((a,b)is an element of Lambda) of a function g is an element of L-2(R-d). We prove that if G(g, Lambda) forms a Schauder basis for L-2(R-d) then the upper Beurling density of Lambda satisfies D+(Lambda) less than or equal to 1. We also prove: that if G(g, Lambda) forms a Schauder basis for L-2(R-d) and if g lies in a the modulation space M-1,M-1(R-d); which is a dense subset of L-2(R-d), or if G(g, Lambda) possesses at least a lower frame bound, then Lambda has uniform Beurling density D(Lambda) = 1. We use related techniques to show that if g is an element of L-1(R-d) boolean AND L-2(R-d) then no collection {g(x - a)}(a is an element of Gamma) of pure translates of g can form a Schauder basis for L-2(R-d). We also extend these results to the case of finitely many generating functions g(1),...,g(r).