A new family of hyperbolic slits in the Gabor frame set of B-spline generators

TL;DR

The paper studies the frame set $\mathcal{F}(N_n)$ of Gabor systems with cardinal B-spline generators $N_n$, proving an infinite family of non-frame obstructions on hyperbolas $ab=p/q<1$. It combines Gröchenig's partition-of-unity obstruction with Nielsen's partly partition-of-unity technique to extend point obstructions (set $P$) into hyperbolic obstructions (set $H$) for all $n$, and situates these within the Janssen tie via the Zak transform and the Zibulski–Zeevi representation. By showing $P \subset H \subset T$ and constructing $q-p+1$ independent vectors in $\ker \Phi^{N_n}(x_0,0)$, it certifies non-frame behavior along the hyperbolic segments, thus revealing a rich, geometry-driven obstruction landscape for $\mathcal{F}(N_n)$. The results connect with known obstructions for $N_1$ and the hat spline $N_2$, and indicate that hyperbolic obstructions persist and contract toward point obstructions as the spline order grows.

Abstract

We exhibit a new infinite family of hyperbolic curves in the complement of the frame set of Gabor systems with B-spline generators. The proof technique is a combination of an approach by Gröchenig [Partitions of unity and new obstructions for Gabor frames, arXiv:1507.08432, 2015] and a partly partition of unity argument by Nielsen and the author [Counterexamples to the B-spline conjecture for Gabor frames, J. Fourier Anal. Appl., 22(6):1440-1451, 2016]. We relate the new hyperbolic obstructions to the "right bow tie" of the so-called Janssen tie [Zak transforms with few zeros and the tie, In Advances in Gabor analysis, Birkhäuser, 2003].

Figures (6)

• Figure 1: The point obstruction set $P$ defined in \ref{['thm:point-obstructions-grochenig']} plotted for $b \le 15$. In each vertical band $a_0 = 1/\mu$, the points become denser and denser as $r$ increases with accumulation point $(1/\mu, \mu)$, $\mu \ge 3$. The colorbar indicates the value of $ab \in \lparen1/2,1\rparen$.
• Figure 2: The new hyperbolic obstruction set $H$ for $N_n$, $n=2$, from \ref{['thm:new-hyperbolas-frame-set-Bn']}. The hyperbolic segments, defined by \ref{['eq:hyperbolic-obstruction-b-interval']}, are colored by their $ab$ values.
• Figure 3: Sketch of the Janssen tie in $b \in \lbrack N, N+1\rbrack$. The shaded region shows the tile $T_N$.
• Figure 4: The obstruction point set $P$ (marked by black dots) and the hyperbolic obstruction set $H$ for $N_n$, $n=2$, with hyperbolic segments colored by their $ab$ values. The shaded regions represent the "right half" of Janssen's tie $T_N$, whose upper and lower boundary curves are shown in dash-dot lines. Note that the plots are restricted to $r \le 50$ to avoid excessive cluttering at the accumulation points.
• Figure 5: The point set $P$ (black dots) and the hyperbolic obstruction set $H$ for $N_2$ (i.e., $n=2$) for large values of $b$. The hyperbolic segments are colored by their $ab$ values. The complexity of the structure of the sets increases as $b$ increases, compared to \ref{['fig:hyperbola-set_zoom']}.

Theorems & Definitions (18)

• Theorem 1.1: Zibulski-Zeevi characterization
• Theorem 1.2: Gröchenig GrochenigPartitions2015
• Theorem 1.3
• Definition 1
• Lemma 2.1
• proof
• Proposition 2.2: Local gaps along $a_0=1/\mu$
• proof
• Corollary 2.3: Gaps above integer $b$ values
• proof