Counterexamples to the B-spline conjecture for Gabor frames

2015/07/14 by Jakob Lemvig, Lemvig, Jakob, Kamilla Haahr Nielsen +1 · 1 citation
Computer Science · Mathematics · Medicine · #42C15 #FOS: Mathematics #Functional Analysis (math.FA) #Image and Signal Denoising Methods #Mathematical Analysis and Transform Methods #Medical Imaging Techniques and Applications #math.FA #msc:42C15

paper · pdf · doi:10.48550/arxiv.1507.03982

Version 2: Lem. 5, Prop. 6, and Thm. 7 added, Version 3: Thm. 8 changed

openalex publication_date 2015/07/14 · arxiv created 2015/08/19 · arxiv updated 2015/08/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

The frame set conjecture for B-splines B_n, n ≥ 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,…, 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 ≥ 2.

Citations

• The abc-problem for Gabor systems

Cited by

• Partitions of Unity and New Obstructions for Gabor Frames

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