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arxiv: 2606.26052 · v1 · pith:5DYMO5ILnew · submitted 2026-06-24 · 🧮 math.FA

On the existence problem of regular Gabor frames

Jaume de Dios Pont , Lukas Liehr , Mitchell A. Taylor This is my paper

Pith reviewed 2026-06-25 19:01 UTC · model grok-4.3

classification 🧮 math.FA
keywords Gabor framesZak transformquasiperiodic functionscommon zeroslattice densityexistence problemSchwartz spaceFeichtinger algebra
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The pith

For d>1, explicit criteria on lattices with density >1 ensure no continuous-Zak function generates a Gabor frame.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that in every dimension d greater than 1, certain lattices in phase space with density exceeding one admit no Gabor frames generated by any window whose Zak transform is continuous. This supplies a negative answer to the existence question for regular Gabor frames with windows from the Schwartz space, the Feichtinger algebra, or the Fourier-invariant Wiener space. The argument proceeds by converting the frame condition into the question of whether a family of quasiperiodic functions shares a common zero. A new algebraic characterization decides exactly when such common zeros occur.

Core claim

For every dimension d>1 there are explicit criteria on lattices Λ subset R^{2d} with D(Λ)>1 such that the Zak transforms associated to any candidate window necessarily share a common zero, so the Gabor system along Λ cannot form a frame.

What carries the argument

Characterization of common zeros for collections of quasiperiodic functions, which reduces the frame existence question to an algebraic condition on the lattice and the window.

If this is right

  • No Gabor frame exists along the identified lattices for any window in the Schwartz space.
  • The same lattices rule out frames generated by windows from the Feichtinger algebra and the Fourier-invariant Wiener space.
  • The common-zero characterization applies directly to other questions about quasiperiodic functions.
  • The negative existence result holds for every d>1 once the lattice satisfies the explicit algebraic criteria.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The lattice geometry alone can force unavoidable zeros in Zak transforms, independent of further regularity assumptions on the window.
  • Similar algebraic obstructions might be checked computationally for concrete lattices in dimensions three and higher.
  • The Lean 4 formalization shows the algebraic criterion is in principle machine-checkable.

Load-bearing premise

A collection of quasiperiodic functions admits a common zero precisely when certain explicit algebraic conditions on the lattice and the functions hold.

What would settle it

Produce a lattice meeting the stated criteria together with a continuous-Zak window whose associated quasiperiodic functions have no common zero on the fundamental domain.

Figures

Figures reproduced from arXiv: 2606.26052 by Jaume de Dios Pont, Lukas Liehr, Mitchell A. Taylor.

Figure 1
Figure 1. Figure 1: Visualization of Theorem 1.3 in the case d = N = 2. The figure shows the heatmap of two quasiperiodic functions s1, s2 : R 4 → C on the intersection of the 4-cube [0, 1]4 with the plane P, parametrized by P : R 2 → R 4 with P(u, v) = (u, v, v, 1 − u). The functions s1 and s2 are chosen as products of the theta function ϑ3 and certain sine functions. The white circle marks a common zero of s1 and s2. We ref… view at source ↗
read the original abstract

For every dimension $d > 1$, we establish explicit criteria on lattices $\Lambda \subset \mathbb{R}^{2d}$ with density $D(\Lambda) > 1$ such that no function with a continuous Zak transform generates a Gabor frame along $\Lambda$. In particular, this gives a negative answer to the existence problem of Gabor frames with window functions in the Schwartz space, the Feichtinger algebra, and the Fourier-invariant Wiener space. Our result is based on a characterization of when a collection of quasiperiodic functions admits a common zero, which may be of independent interest. We also include a formalization of our main result in Lean 4.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper establishes explicit criteria on lattices Λ ⊂ ℝ^{2d} (d > 1) with density D(Λ) > 1 such that no function possessing a continuous Zak transform generates a Gabor frame along Λ. This yields negative answers to the existence of regular Gabor frames with windows in the Schwartz space, Feichtinger algebra, and Fourier-invariant Wiener space. The argument rests on a new algebraic characterization of when collections of quasiperiodic functions admit a common zero; the main result is formalized in Lean 4.

Significance. If the result holds, it supplies concrete obstructions that resolve an open existence question in time-frequency analysis for highly regular windows on overcomplete lattices. The characterization of common zeros for quasiperiodic functions is of independent interest, and the Lean 4 formalization provides machine-checked verification of the central derivation, strengthening the reliability of the algebraic reduction from the frame condition to the zero-set condition.

minor comments (2)
  1. The abstract and introduction would benefit from a brief statement of the precise form of the explicit criteria on the lattice (e.g., the algebraic conditions on the generators of Λ) to allow readers to assess applicability without reading the full characterization section.
  2. Notation for the Zak transform and the quasiperiodic functions could be introduced with a short reminder of their periodicity properties in §2 to improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report, the recognition of the significance of our results, and the recommendation to accept the manuscript. We appreciate the comments on the independent interest of the algebraic characterization and the Lean 4 formalization.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via machine-checked characterization

full rationale

The paper's central step converts the Gabor frame question into a common-zero question for quasiperiodic functions via an explicit algebraic characterization on the lattice. This characterization is formalized and verified in Lean 4, supplying direct machine-checked support independent of the present paper's fitted values or self-referential definitions. No load-bearing self-citation chains, ansatzes smuggled via prior work, or predictions that reduce to inputs by construction are present. The result is externally falsifiable via the formalization and applies to standard function spaces without redefining its own premises.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on a new characterization of common zeros of quasiperiodic functions together with standard facts from Gabor analysis and lattice theory; no free parameters, ad-hoc axioms, or invented entities are indicated.

axioms (1)
  • standard math Standard properties of the Zak transform and Gabor systems in L2(R^d) as developed in prior literature.
    Invoked implicitly when translating the frame condition into a zero-set condition.

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Reference graph

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