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

TL;DR

This paper identifies a new family of hyperbolic curves that serve as obstructions in the Gabor frame set for B-spline generators, expanding understanding of frame conditions through advanced mathematical techniques.

Contribution

It introduces a novel infinite family of hyperbolic obstructions in the Gabor frame set for B-splines, combining existing analytical approaches.

Findings

New hyperbolic obstructions identified

Relation to Janssen tie and Zak transforms established

Advances understanding of Gabor frame set boundaries

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 Gr\"ochenig [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, Birkh\"auser, 2003].