Counterexamples to the B-spline Conjecture for Gabor Frames

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 \(b=2,3,\dots \), not belonging to the frame set of the nth 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 \(B_n\), \(n \ge 2\).