Obstructions for Gabor frames of the second-order B-spline

For a window \( g\in L^2(\mathbb {R}) \), the subset of all lattice parameters \( (a, b)\in \mathbb {R}^2_+ \) such that \( \mathcal {G}(g,a,b)=\{e^{2\pi ib m\cdot }g(\cdot -a k): k, m\in \mathbb {Z}\} \) forms a frame for \( L^2(\mathbb {R}) \) is known as the frame set of g. In time-frequency analysis, determining the Gabor frame set for a given window is a challenging open problem. In particular, the frame set for B-splines has many obstructions. Lemvig and Nielsen in (J. Fourier Anal. Appl. 22, 1440–1451, 2016) conjectured that if

$$\begin{aligned} a_0=\dfrac{1}{2m+1},~ b_0=\dfrac{2k+1}{2},~k,m\in \mathbb {N},~k>m,~a_0b_0<1, \end{aligned}$$

then the Gabor system \( \mathcal {G}(Q_2, a, b) \) of the second-order B-spline \( Q_2 \) is not a frame along the hyperbolas

$$\begin{aligned} ab=\dfrac{2k+1}{2(2m+1)},\text { for }b\in \left[ b_0-a_0\dfrac{k-m}{2}, b_0+a_0\dfrac{k-m}{2}\right] , \end{aligned}$$

for every \( a_0 \), \( b_0 \). Nielsen in (2015) also conjectured that \( \mathcal {G}(Q_2, a,b) \) is not a frame for

$$a=\dfrac{1}{2m},~b=\dfrac{2k+1}{2},~k,m\in \mathbb {N},~k>m,~ab<1\text { with }\gcd (4m,2k+1)=1.$$

In this paper, we prove that both Conjectures are true.