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 $$\mathcal{G}(g;\alpha,\beta):=\{g_{m,n}\}_{m,n\in\mathbb{Z}}=\{e^{2\pi i \beta m x}g(x-\alpha n)\}_{m,n\in\mathbb{Z}}$$ always forms a frame in $L^2(\mathbb{R})$ for all lattice parameters $\alpha$,$\beta$, $\alpha\beta\leq 1$.