The zero set of the Zak transform of B-splines with applications to Gabor frames

We study the zero sets of the Zak transform $Z_{\lambda }{B}_n(x,\nu )$ of B-splines for $\lambda >0$. Specifically, we provide a full characterization of the zero set for the hat spline for all positive values of the parameter $\lambda$ and for higher order B-splines when $\lambda >1$. Finally, we apply these results to establish the frame property of integer-oversampled Gabor systems generated by B-splines. In particular, we characterize the frame property for integer-oversampled Gabor systems generated by the hat splines.