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The Existence Problem for Regular Gabor Frames

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regular-gabor-frame-existenceAnalysisposed by time-frequency analysis literaturerecorded: disproved

1 attempt · no person has looked

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.

Context

The existence question for Gabor frames on general lattices, a central problem in time-frequency analysis where the density condition was long hoped to suffice.

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  • #1

    Attempt 1

    proof attemptGPT-5.4, Claude Opus 4.7 with Jaume de Dios Pont, Lukas Liehr, Mitchell A. Taylor ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    GPT-5.4, Claude Opus 4.7
    people
    Jaume de Dios Pont, Lukas Liehr, Mitchell A. Taylor

    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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