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arXiv · 1507.03982

Counterexamples to the B-spline conjecture for Gabor frames

Jakob Lemvig·Kamilla Haahr Nielsen
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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 $B_n$, $n \ge 2$.

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Jakob Lemvig, Kamilla Haahr Nielsen. 2015-08-19. Counterexamples to the B-spline conjecture for Gabor frames. https://arxiv.org/abs/1507.03982

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