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The Existence Problem for Regular Gabor Frames

Analysis · posed by time-frequency analysis literature · disproved

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Statement

Does every lattice of density above one admit a Gabor frame with a nice window? No. For every dimension d>1d > 1 there are explicit criteria on lattices Λ⊂R2d\Lambda \subset \mathbb{R}^{2d} with D(Λ)>1D(\Lambda) > 1 such that no function with continuous Zak transform generates a Gabor frame along Λ\Lambda, which answers the existence problem negatively for Schwartz-class windows.

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. proof attempt · #1

    Jaume de Dios Pont, Lukas Liehr and Mitchell A. Taylor, using GPT-5.4, Claude Opus 4.7

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

    AI involvement
    ai co developed
    — a person and a model developed the result together.

    The disclosure separates the two roles: GPT-5.4 was used mainly for mathematical exploration, including exploring whether homology-theoretic methods could give a common-zero criterion for two quasiperiodic functions, while Claude Opus 4.7 assisted with the Lean formalization. The authors verified everything independently. The same group's Cantor Fourier frame paper is also in this catalog.

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