On Gabor frames generated by sign-changing windows and B-splines

TL;DR

This paper characterizes when Gabor systems with certain compactly supported windows form frames, identifying obstructions on specific curves and confirming a conjecture for B-splines within a particular parameter region.

Contribution

It provides a detailed characterization of the frame property for Gabor systems with sign-changing windows and confirms a conjecture about B-splines in a specific parameter region.

Findings

Obstructions to the frame property are on countable curves.

For positive interior functions, the entire region is in the frame set.

Confirmed the B-spline conjecture in a specific parameter region.

Abstract

For a class of compactly supported windows we characterize the frame property for a Gabor system $\mts,$ for translation parameters $a$ belonging to a certain range depending on the support size. We show that the obstructions to the frame property are located on a countable number of "curves." For functions that are positive on the interior of the support these obstructions do not appear, and the considered region in the $(a,b)$ plane is fully contained in the frame set. In particular this confirms a recent conjecture about B-splines by Gr\"ochenig in that particular region. We prove that the full conjecture is true if it can be proved in a certain "hyperbolic strip."