Obstructions for Gabor frames of the second order B-spline

TL;DR

This paper proves two conjectures that identify specific obstructions preventing Gabor systems of the second order B-spline from forming frames, advancing understanding of the frame set in time-frequency analysis.

Contribution

It confirms two conjectures regarding obstructions in the Gabor frame set for second order B-splines, clarifying when such systems are not frames.

Findings

Both conjectures are proven true.

Identifies specific hyperbolas where Gabor systems are not frames.

Provides conditions under which Gabor frames of second order B-splines fail.

Abstract

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 \cite{counter} conjectured that if \begin{align} a_0=\dfrac{1}{2m+1},~ b_0=\dfrac{2k+1}{2},~k,m\in \mathbb{N},~k>m,~a_0b_0<1,\nonumber \end{align} 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{align} 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],\nonumber \end{align} for every $a_0$, $b_0$. Nielsen in \cite {Nielsenthesis} 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.