The frameset of the second-order B-Spline along $ab=1/q$

Abstract

We give a complete characterization of the zero set of the Zak transform of the second-order B-spline $Q_2$. As a consequence, we obtain a new proof of the Lemvig--Nielsen obstructions to the Gabor frame property of $Q_2$, and, along the hyperbola $ab=1/q$, we show these obstructions are sharp: every point they do not exclude belongs to the frame set. This gives the first complete characterization of $\mathcal{F}(Q_2)$ along an entire curve in this family, resolving a question left open by Lemvig and Nielsen.