Counterexamples to the B-spline conjecture for Gabor frames

Abstract

The frame set conjecture for B-splines $B_n$, $n\ge 2$, states that the frame set is the maximal set that avoids the known obstructions. We show that any hyperbola of the form $ab=r$, where $r$ is a rational number smaller than one and $a$ and $b$ denote the sampling and modulation rates, respectively, has infinitely many pieces, located around $b=2,3,\dots$, \emph{not} belonging to the frame set of the $n$th order B-spline. This, in turn, disproves the frame set conjecture for B-splines. On the other hand, we uncover a new region belonging to the frame set for B-splines $B_n$, $n\ge 2$.

Version 2: Lem. 5, Prop. 6, and Thm. 7 added, Version 3: Thm. 8 changed

References in corpus (2)

• Zak transforms and Gabor frames of totally positive functions and exponential B-splines
• The $abc$-problem for Gabor systems

Cited by in corpus (1)

• Partitions of Unity and New Obstructions for Gabor Frames