Gabor frames for functions supported on a semi-axis

Let $g\in L^2(\mathbb{R})$ be a strictly decreasing continuous function supported on ${\mathbb{R}}_{+}$ such that for all $t>0$ we have $g(x+t)\le q(t)g(x)$ for some $q(t)<1$. We prove that the Gabor system $$\mathcal{G}(g;\alpha ,\beta ):=\{g_{m,n}{\}}_{m,n\in \mathbb{Z}}=\{e^{2\pi i\beta mx}g(x-\alpha n){\}}_{m,n\in \mathbb{Z}}$$ always forms a frame in $L^2(\mathbb{R})$ for all lattice parameters $\alpha$,$\beta$, $\alpha \beta \le 1$.