7 research outputs found
Counterexamples to the B-spline conjecture for Gabor frames
The frame set conjecture for B-splines , , states that the
frame set is the maximal set that avoids the known obstructions. We show that
any hyperbola of the form , where is a rational number smaller than
one and and denote the sampling and modulation rates, respectively, has
infinitely many pieces, located around , \emph{not} belonging to
the frame set of the 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 , .Comment: Version 2: Lem. 5, Prop. 6, and Thm. 7 added, Version 3: Thm. 8
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