  The frame set of a window $\phi\in L^2(\mathbb{R})$ is the subset of all
lattice parameters $(\alpha, \beta)\in \mathbb{R}^2_+$ such that
$\mathcal{G}(\phi,\alpha,\beta)=\{e^{2\pi i\beta m\cdot}\phi(\cdot-\alpha k) :
k, m\in\mathbb{Z}\}$ forms a frame for $L^2(\mathbb{R})$. In this paper, we
investigate the frame set of B-splines, totally positive functions, and Hermite
functions. We derive a sufficient condition for Gabor frames using the
connection between sampling theory in shift-invariant spaces and Gabor
analysis. As a consequence, we obtain a new frame region belonging to the frame
set of B-splines and Hermite functions. For a class of functions that includes
certain totally positive functions, we prove that for any choice of lattice
parameters $\alpha, \beta>0$ with $\alpha\beta<1,$ there exists a $\gamma>0$
depending on $\alpha\beta$ such that
$\mathcal{G}(\phi(\gamma\cdot),\alpha,\beta)$ forms a frame for
$L^2(\mathbb{R})$.
