Counterexamples to the B-spline Conjecture for Gabor Frames
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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 \(B_n\), \(n \ge 2\).
KeywordsB-spline Frame Frame set Gabor system Zibulski–Zeevi matrix
Mathematics Subject ClassificationPrimary 42C15 Secondary 42A60
The authors would like to thank Ole Christensen for posing the problem of characterizing the frame set of B-splines in a talk at the Technical University of Denmark on February 3, 2015, that initiated the work presented in this paper. The authors would also like to thank Karlheinz Gröchenig and the referees for comments improving the presentation of the paper.
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