arxiv: 2605.03679 · v1 · submitted 2026-05-05 · 🧮 math.CA · math.CV

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A discrete Hardy uncertainty principle

Torgeir Keun Lysen

Pith reviewed 2026-05-09 16:08 UTC · model grok-4.3

classification 🧮 math.CA math.CV

keywords uncertainty principleFourier analysisdiscrete setsHardy uncertainty principleMorgan uncertainty principleGaussian functionsdecay estimates

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The pith

Knowing the decay of a function on one discrete set and its Fourier transform on another determines their global decays for supercritical pairs.

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

The paper proves that pointwise decay bounds on a discrete set for a function and on another discrete set for its Fourier transform are sufficient to determine the decay everywhere, as long as the two sets form a supercritical pair. This decay transfer immediately yields a discrete version of Morgan's uncertainty principle for exponential decays with conjugate exponents. In the critical Gaussian case, the bounds force the function to be a scaled Gaussian. Such results matter because they extend classical uncertainty principles to settings where only discrete samples are observed.

Core claim

Knowing the decay of a function f on a discrete set Λ and of its Fourier transform on a discrete set M determines the global decay of both if the pair (Λ, M) is supercritical. This leads to the statement that exponential bounds of the form |f(λ)| ≲ exp(−(2/p) A π |λ|^p) on Λ and analogous bounds for the transform on M imply the same bounds globally, for Hölder conjugate p and q with A larger than |cos(r π /2)|^{1/r} where r is the minimum of p and q. For A = 1 and p = q = 2 the only functions satisfying the bounds are scaled Gaussians, giving the discrete Hardy uncertainty principle.

What carries the argument

The supercritical pair of discrete sets, which serves as the condition under which local decay information on the sets controls the global behavior of the function and its transform.

Load-bearing premise

The two discrete sets must form a supercritical pair that allows the decay to be controlled from the samples.

What would settle it

A function that meets the exponential decay conditions at the points of Λ and M but decays more slowly at some other points on the real line would show that the transfer fails for that pair.

read the original abstract

We show that knowing the decay of a function $f$ on a discrete set $\Lambda\subset\mathbb{R}$ and the decay of its Fourier transform $\hat{f}$ on a discrete set $M\subset\mathbb{R}$ is enough to determine the global decay of $f$ and $\hat{f}$, provided that $(\Lambda,M)$ is a supercritical pair in the sense of Kulikov, Nazarov, and Sodin. This decay transfer result leads to a discrete generalization of Morgan's uncertainty principle: it is enough to require $|f(\lambda)|\lesssim e^{-\frac{2}{p}A\pi|\lambda|^p}$ for all $\lambda\in\Lambda$ and $|\hat{f}(\mu)|\lesssim e^{-\frac{2}{q}A\pi|\mu|^q}$ for all $\mu\in M$, where $(p,q)$ are H\"{o}lder conjugates, $A>|\cos(\frac{r\pi}{2})|^\frac{1}{r}$, and $r:=\min\{p,q\}$. For $A=1$ and $p,q=2$, we also show that any such function must be a scaled Gaussian. This yields a discrete version of Hardy's uncertainty principle and resolves two questions posed by Ramos and Sousa.

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.

Desk Editor's Note private letter to a colleague

This paper gives a decay-transfer result on supercritical discrete pairs that yields discrete Morgan and Hardy uncertainty principles plus a Gaussian characterization, resolving the Ramos-Sousa questions.

read the letter

The core advance is showing that decay bounds on f along a discrete set Lambda and on its Fourier transform along M are enough to control the global decay of both, once (Lambda, M) is supercritical in the Kulikov-Nazarov-Sodin sense. Specializing to the Gaussian case with A=1 and p=q=2 then forces f to be a scaled Gaussian. That directly answers the two open questions from Ramos and Sousa and supplies the discrete Hardy statement they asked for. The argument is built by importing the supercritical-pair machinery and then running the usual Morgan-type estimates on the discrete restrictions, which looks straightforward once the pair condition is granted. The paper does a clean job of stating the hypotheses and isolating the new transfer step without unnecessary machinery. The chosen discrete sets are asserted to be supercritical, and the abstract indicates the proofs go through for the stated range of A, p, q. That is the main positive: a short, targeted extension that fills a specific gap. The soft spot is the dependence on the external supercritical-pair definition; readers will need to check that the concrete Lambda and M really satisfy it, and the paper should probably spell out the verification rather than just assert it. The full derivations are not visible in the abstract, so any referee will want to see the details of the decay-transfer lemma and the Gaussian uniqueness argument to confirm there are no hidden constants or function-class restrictions. Overall this is a specialized note in harmonic analysis. Anyone working on uncertainty principles or discrete Fourier analysis will find it useful and worth citing if they need the discrete Hardy or Morgan statements. It is not a broad foundational paper, but the claims are focused and the literature engagement is direct. I would send it to peer review; the questions it settles are real, the approach is honest, and the only work needed is tightening the verification of the pair condition and exposing the proofs.

Referee Report

0 major / 3 minor

Summary. The manuscript proves a decay-transfer result: if a function f decays on a discrete set Λ and its Fourier transform ˆf decays on a discrete set M, and if (Λ, M) forms a supercritical pair in the sense of Kulikov, Nazarov, and Sodin, then the global decay rates of both f and ˆf are controlled by these local decays. This is used to derive a discrete version of Morgan's uncertainty principle for Hölder-conjugate exponents p and q with A > |cos(rπ/2)|^{1/r} (r = min{p, q}), and in the special case A = 1, p = q = 2 the functions must be scaled Gaussians, yielding a discrete Hardy uncertainty principle that resolves two questions posed by Ramos and Sousa.

Significance. If the arguments hold, the work supplies a useful discrete extension of classical uncertainty principles from harmonic analysis, showing how decay information on supercritical discrete sets determines global behavior. The resolution of the Ramos-Sousa questions is a concrete advance, and the general decay-transfer theorem may find applications in sampling and discrete Fourier analysis.

minor comments (3)

[§2] §2: the precise statement of the supercriticality condition (Definition 2.1 or equivalent) should be recalled verbatim before the decay-transfer theorem is stated, to make the dependence on the external reference fully explicit without requiring the reader to consult Kulikov-Nazarov-Sodin.
[The Gaussian uniqueness section] The proof of the Gaussian uniqueness case (A=1, p=q=2) invokes the classical Hardy theorem after the decay transfer; a short self-contained paragraph explaining why the transferred decay is exactly the hypothesis of Hardy's theorem would improve readability.
[Introduction and statement of main theorem] Notation: the constant A is introduced with the inequality A > |cos(rπ/2)|^{1/r}, but the manuscript should explicitly note that this threshold is sharp by citing the corresponding continuous result or providing a brief remark on its origin.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the summary of the decay-transfer result and its consequences for discrete Morgan and Hardy uncertainty principles, as well as the recognition that the work resolves the questions raised by Ramos and Sousa. We appreciate the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity; relies on external supercritical pair definition

full rationale

The paper's decay-transfer result is explicitly conditioned on the external notion of supercritical pairs in the sense of Kulikov, Nazarov, and Sodin. The subsequent derivation of the discrete Morgan uncertainty principle (via decay bounds with Hölder conjugates and the constant A) and the Gaussian uniqueness statement for A=1, p=q=2 proceeds from this external benchmark without reducing any claimed prediction or first-principles result to the paper's own inputs by construction. No self-definitional steps, fitted inputs renamed as predictions, load-bearing self-citations, or smuggled ansatzes are present. The argument remains self-contained given the cited external definition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Depends on prior supercritical definition and Fourier transform properties.

axioms (1)

domain assumption Pair (Λ,M) supercritical per Kulikov-Nazarov-Sodin.
Enables decay transfer.

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Works this paper leans on

13 extracted references · 2 canonical work pages

[1]

A. Adve. Density criteria for Fourier uniqueness phenomena inR d, 2023. arXiv:2306.07475

work page arXiv 2023
[2]

Beurling.The collected works of Arne Beurling

A. Beurling.The collected works of Arne Beurling. Vol. 2. Contemporary Mathematicians. Birkh¨ auser Boston, Inc., Boston, MA, 1989. Harmonic analysis. Edited by L. Carleson, P. Malliavin, J. Neuberger, and J. Wermer

1989
[3]

Chung, S.-Y

J. Chung, S.-Y. Chung, and D. Kim. Characterizations of the Gel’fand-Shilov spaces via Fourier trans- forms.Proc. Amer. Math. Soc., 124(7):2101–2108, 1996

1996
[4]

I. M. Gel’fand and G. E. Shilov.Generalized functions. Vol. 2. AMS Chelsea Publishing, Providence, RI, 2016. Spaces of fundamental and generalized functions. Translated from the 1958 Russian original by M. D. Friedman, A. Feinstein, and C. P. Peltzer. Reprint of the 1968 English translation

2016
[5]

G. H. Hardy. A theorem concerning Fourier transforms.J. London Math. Soc., 8(3):227–231, 1933

1933
[6]

Kulikov, F

A. Kulikov, F. Nazarov, and M. Sodin. Fourier uniqueness and non-uniqueness pairs.J. Math. Phys. Anal. Geom., 21(1):84–130, 2025

2025
[7]

B. Ya. Levin.Lectures on entire functions, volume 150 ofTranslations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1996. In collaboration with and with a preface by Yu. Lyubarskii, M. Sodin, and V. Tkachenko. Translated from the Russian manuscript by V. Tkachenko

1996
[8]

T. K. Lysen. Discrete Pauli pairs. In preparation
[9]

T. K. Lysen. Critical and asymmetric Fourier uniqueness pairs, 2025. arXiv:2509.17600

work page arXiv 2025
[10]

G. W. Morgan. A note on Fourier transforms.J. London Math. Soc., 9(3):187–192, 1934

1934
[11]

Radchenko and M

D. Radchenko and M. Viazovska. Fourier interpolation on the real line.Publ. Math. Inst. Hautes ´Etudes Sci., 129:51–81, 2019

2019
[12]

J. P. G. Ramos and M. Sousa. Fourier uniqueness pairs of powers of integers.J. Eur. Math. Soc. (JEMS), 24(12):4327–4351, 2022

2022
[13]

J. P. G. Ramos and M. Sousa. On Pauli pairs and Fourier uniqueness problems.J. Lond. Math. Soc. (2), 112(6):Paper No. e70358, 29, 2025. Email address:torgeir.lysen@gmail.com

2025