Frames for compactly supported functions with irrational density

We find sufficient conditions on a compactly supported function g, supp g = [a,b], which guarantee that the Gabor system align* &G(g;α,β)=2π i β m xg(x-α n),nalign* is a frame for all α < b-a, α β < 1, α β ∉ Q. These conditions are on one hand satisfied by almost all such functions, and on the other hand are explicit enough that we can give many concrete examples of the functions g which give us a frame, for example, g(x) = exp (1x⁴-1)χ (-1,1)(x).