We find sufficient conditions on a compactly supported function $g$, $\supp g = [a,b]$ which guarantee that the Gabor system $$\mathcal{ G}(g;\alpha,\beta)=\{ e^{ 2\pi i \beta m x}g(x-\alpha n)\}_{ m,n\in\mathbb{ Z}}$$ is a frame for all $\alpha < b-a, \alpha\beta < 1, \alpha\beta \notin\Q$. 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 e.g. $g(x) = \exp(\frac{ 1}{ x^4-1})\chi_{ (-1,1)}(x)$.
Categories: math.FA