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arxiv: 2605.26709 · v1 · pith:BYBOG445new · submitted 2026-05-26 · 🧮 math.CA · math.FA

On a fundamental barrier of the Wirtinger criterion for Gabor systems with odd functions

Markus Faulhuber This is my paper

Pith reviewed 2026-07-01 16:04 UTC · model grok-4.3

classification 🧮 math.CA math.FA
keywords Gabor systemsWirtinger criterionframe setHermite functionsodd functionsdensityGabor framestime-frequency analysis
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The pith

The Wirtinger criterion cannot be used to investigate frame sets of density less than 2 for odd functions in Gabor systems.

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

The paper establishes that the Wirtinger criterion has a fundamental barrier when the window function is odd. This barrier prevents it from determining whether Gabor systems form frames in regions where the density is less than 2. As a result, the criterion cannot be applied to the frame set conjecture for the first Hermite function. A reader would care because it shows that this particular tool is ineffective for a class of problems in time-frequency analysis involving odd functions.

Core claim

We show that the Wirtinger criterion cannot be used to investigate the frame set conjecture for the first Hermite function. More generally, for odd functions, it cannot determine regions of the frame set with density less than 2.

What carries the argument

The Wirtinger criterion applied to Gabor systems generated by odd functions, which encounters an obstruction at densities below 2 due to parity.

If this is right

  • The Wirtinger criterion is ruled out as a method for studying the frame set of the first Hermite function.
  • For odd window functions, the criterion yields no information on frame sets with density less than 2.
  • Alternative approaches are required to explore the frame set conjecture in the presence of odd functions.

Where Pith is reading between the lines

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

  • Parity of the window function appears to be a critical factor limiting the applicability of the Wirtinger criterion.
  • Similar barriers might exist for other criteria when functions have specific symmetry properties.
  • Investigations into frame sets for odd functions will likely require tools that do not rely on the Wirtinger approach.

Load-bearing premise

The barrier arises specifically from the odd parity of the functions preventing the Wirtinger criterion from providing information below density 2.

What would settle it

A successful use of the Wirtinger criterion to characterize a frame set region with density less than 2 for an odd function would falsify the result.

read the original abstract

We show that the Wirtinger criterion cannot be used to investigate the frame set conjecture for the first Hermite function. More generally, for odd functions, it cannot determine regions of the frame set with density less than 2.

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 / 1 minor

Summary. The manuscript shows that the Wirtinger criterion cannot be used to investigate the frame set conjecture for the first Hermite function. More generally, for odd functions, it cannot determine regions of the frame set with density less than 2.

Significance. If the result holds, it identifies a fundamental limitation of the Wirtinger criterion arising directly from the odd parity of the generator: the relevant integrals or inner products vanish or cancel by symmetry for densities below 2. This negative result is useful for the field, as it explains why the criterion is inapplicable to the first Hermite function (a standard test case) and directs attention to other methods for low-density Gabor frames.

minor comments (1)
  1. The manuscript is concise; a short paragraph recalling the precise statement of the Wirtinger criterion (including the integrals involved) would improve accessibility without lengthening the note substantially.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report and the recommendation to accept the manuscript. The referee's summary accurately captures the contribution: the Wirtinger criterion is inapplicable to the frame set conjecture for the first Hermite function and, more generally, cannot determine frame regions for odd generators at densities below 2.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper establishes a limitation of the Wirtinger criterion for odd functions by direct appeal to parity-induced cancellations in the relevant integrals or inner products, which force the criterion to be uninformative below density 2. This follows from the definitions of oddness and the criterion itself, without any reduction of a claimed prediction to a fitted input, self-definitional loop, or load-bearing self-citation. The argument is a straightforward symmetry obstruction and remains self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.1-grok · 5547 in / 1043 out tokens · 60842 ms · 2026-07-01T16:04:46.208904+00:00 · methodology

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

  1. [1]

    Christensen.An Introduction to Frames and Riesz Bases

    O. Christensen.An Introduction to Frames and Riesz Bases. Applied and Numerical Harmonic Analysis. Birkh¨ auser, 2. edition, 2016. 10.1007/978-3-319-25613-9

    work page doi:10.1007/978-3-319-25613-9 2016

  2. [2]

    Faulhuber

    M. Faulhuber. A short note on the frame set of odd functions.Bulletin of the Australian Mathematical Society, 98(3):481–493, 2018. 10.1017/S0004972718000746. 6 M. FAULHUBER

    work page doi:10.1017/s0004972718000746 2018

  3. [3]

    Faulhuber

    M. Faulhuber. On the Parity under Metaplectic Operators and an Extension of a Result of Lyubarskii and Nes.Results in Mathematics, 75, 2020. 10.1007/s00025-019-1134-4

    work page doi:10.1007/s00025-019-1134-4 2020

  4. [4]

    Faulhuber

    M. Faulhuber. Gabor systems with Hermite functions of ordernand oversampling greater thann+1 which are not frames.Sampling Theory, Signal Processing, and Data Analysis, 23:22, 2025. 10.1007/s43670-025-00112-5

    work page doi:10.1007/s43670-025-00112-5 2025

  5. [5]

    Faulhuber

    M. Faulhuber. Operator-isomorphism pairs and Zak transform methods for the study of Gabor systems. InProceedings of the International Conference on Sampling Theory and Applications, 2025. 10.1109/SampTA64769.2025.11133554

    work page doi:10.1109/sampta64769.2025.11133554 2025

  6. [6]

    Faulhuber, I

    M. Faulhuber, I. Shafkulovska, and I. Zlotnikov. On the frame property of Hermite func- tions and explorations of their frame sets.Journal of Fourier Analysis and Applications, 31(21), 2025. 10.1007/s00041-025-10153-y

    work page doi:10.1007/s00041-025-10153-y 2025

  7. [7]

    G. B. Folland.Harmonic Analysis in Phase Space. Number 122 in Annals of Mathematics Studies. Princeton University Press, 1989. 10.1515/9781400882427

    work page doi:10.1515/9781400882427 1989

  8. [8]

    Ghosh and A

    R. Ghosh and A. A. Selvan. On Gabor frames generated by B-splines, totally posi- tive functions, and Hermite functions.Applied Numerical Mathematics, 207:1–23, 2025. 10.1016/j.apnum.2024.08.021

    work page doi:10.1016/j.apnum.2024.08.021 2025

  9. [9]

    M. A. de Gosson.Symplectic Methods in Harmonic Analysis and in Mathemat- ical Physics, volume 7 ofPseudo-Differential Operators. Theory and Applications. Birkh¨ auser/Springer Basel AG, Basel, 2011. 10.1007/978-3-7643-9992-4

    work page doi:10.1007/978-3-7643-9992-4 2011

  10. [10]

    M. A. de Gosson. Hamiltonian deformations of Gabor frames: First steps.Applied and Computational Harmonic Analysis, 38(2):196–221, 2015. 10.1016/j.acha.2014.03.010

    work page doi:10.1016/j.acha.2014.03.010 2015

  11. [11]

    Gr¨ ochenig.Foundations of Time-Frequency Analysis

    K. Gr¨ ochenig.Foundations of Time-Frequency Analysis. Applied and Numerical Har- monic Analysis. Birkh¨ auser, Boston, MA, 2001. 10.1007/978-1-4612-0003-1

    work page doi:10.1007/978-1-4612-0003-1 2001

  12. [12]

    Gr¨ ochenig

    K. Gr¨ ochenig. The Mystery of Gabor Frames.Journal of Fourier Analysis and Applica- tions, 20(4):865–895, 2014. 10.1007/s00041-014-9336-3

    work page doi:10.1007/s00041-014-9336-3 2014

  13. [13]

    Gr¨ ochenig and Y

    K. Gr¨ ochenig and Y. Lyubarskii. Gabor frames with Hermite functions.Comptes Rendus Mathematique, 344(3):157–162, 2007. 10.1016/j.crma.2006.12.013

    work page doi:10.1016/j.crma.2006.12.013 2007

  14. [14]

    Gr¨ ochenig and Y

    K. Gr¨ ochenig and Y. Lyubarskii. Gabor (super)frames with Hermite functions.Mathe- matische Annalen, 345(2):267–286, 2009. 10.1007/s00208-009-0350-8

    work page doi:10.1007/s00208-009-0350-8 2009

  15. [15]

    Gr¨ ochenig, J

    K. Gr¨ ochenig, J. L. Romero, and J. St¨ ockler. Sampling theorems for shift-invariant spaces, Gabor frames, and totally positive functions.Inventiones mathematicae, 211(3):1119– 1148, 2018. 10.1007/s00222-017-0760-2

    work page doi:10.1007/s00222-017-0760-2 2018

  16. [16]

    Lyubarskii and P

    Y. Lyubarskii and P. G. Nes. Gabor frames with rational density.Applied and Compu- tational Harmonic Analysis, 34(3):488–494, 2013. 10.1016/j.acha.2012.09.001

    work page doi:10.1016/j.acha.2012.09.001 2013

  17. [17]

    Sun and X

    W. Sun and X. Zhou. Irregular Gabor frames and their stability.Proceedings of the American Mathematical Society, 131:2883–2893, 2003. 10.1090/S0002-9939-02-06931-9

    work page doi:10.1090/s0002-9939-02-06931-9 2003