Extremizers of a Fourier uncertainty principle related to averaging

Miquel Saucedo; Sergey Tikhonov

Extremizers of a Fourier uncertainty principle related to averaging

T0 review · 0 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper proves that the averaging uncertainty inequality has a sharp positive constant and an extremizer for every $\alpha,\beta>0$, and finds explicit extremizers, including the interval, in four regimes.

desk verdict Solid paper: proves the general positivity/existence result and confirms Steinerberger's conjecture; the one flagged sign inequality is actually fine, with just a minor typo in a coefficient definition. read the letter →

arxiv 2508.17938 v1 pith:QTDLDIAH submitted 2025-08-25 math.CA

Miquel Saucedo , Sergey Tikhonov This is my paper · ORCID

classification math.CA MSC 42B1042B35

keywords Fourieruncertaintyprincipleextremizersoptimalaveragingradialmeasuressharpconstantstotalvariationconvexdualitytrigonometricinequalities

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies the averaging uncertainty inequality $\|\hat{\mu}(\xi)|\xi|^\beta\|_\infty^\alpha (\int |x|^\alpha\,d\mu)^\beta \ge C \|\mu\|_{TV}^{\alpha+\beta}$ for finite non-negative measures, which arose from an axiomatic approach to optimal averaging. The first main result is that the sharp constant $C_{\alpha,\beta,d}$ is strictly positive for every $\alpha,\beta>0$, and that a non-zero radial extremizer exists in every dimension. The second main result is a structural description of all radial extremizers: either their weighted Fourier transform reaches its supremum at arbitrarily large frequencies, or the measure is a finite sum of spherical-shell measures. In four concrete regimes the extremizers are identified explicitly — the interval characteristic function for $d=1$, $beta=1$, $\alpha\ge 2$ (confirming the conjecture that the interval gives the smoothest average), the tent function for $(\alpha,\beta,d)=(1,2,1)$, a radial $|x|^{-1}$ density in $\mathbb{R}^3$, and the uniform sphere measure in $\mathbb{R}^3$ — with exact sharp constants. A companion corollary fixes the decay rate of the Fourier transform of every extremizer, which was previously open.

What carries the argument

Lemma 3.2 is the engine of the paper. For a radial candidate $\mu_*$ it asks for constants $C>0$, $D\in\mathbb{R}$ and a function $\psi$ given by a sum of Fourier transforms of spherical measures, supported on the frequency set where $\hat\mu_*|\xi|^\beta$ is extremal, such that $H=\psi+C|x|^\alpha-D$ is non-negative and vanishes on the support of $\mu_*$. When this holds, a Plancherel step shows that every competitor with equal total mass and equal weighted Fourier supremum has $I_\mu(|x|^\alpha)\ge I_{\mu_*}(|x|^\alpha)$, so $\mu_*$ is extremal; the same lemma also yields the uniqueness criteria. Section 4 upgrades the idea to a genuine Fenchel–Rockafellar duality for the variational problem, with slackness conditions that force the support of the dual variable onto the extremal frequency set. The dichotomy of Theorem 1.5 and the decay corollary both follow from this duality plus the classical asymptotics of Bessel functions.

What would settle it

Evaluate numerically the coefficients $c_k^{(\alpha)}=\int_0^{1/2}(1-|2x|^\alpha)\cos(2\pi(k+\tfrac12)x)\,dx$ for, say, $\alpha=3,4,5$ and $k=0,1,2,\dots$; if any $k$ yields $c_k^{(\alpha)}(-1)^k(k+\frac12)^3\le 0$, then the certificate $H$ in the interval case fails to be non-negative and the extremality proof via Lemma 3.2 collapses. A direct check of the positivity of $H(x)=|2x|^\alpha+|2(x-1)|^\alpha-2$ on $(1/2,3/2]$ for large $\alpha$ would independently test the claim.

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Extended reading notes

Core claim

The central claim is that the averaging uncertainty principle is sharp in every parameter range and that its extremizers are not merely abstract: they obey a clean dichotomy and can be written down for several natural parameter choices. For radial measures the dichotomy states that an extremizer either has the property that $|\hat\mu(\xi)||\xi|^\beta$ attains its essential supremum for arbitrarily large $|\xi|$, or it is a finite non-negative combination of uniform spherical-shell measures. A certificate function $H(x)=\psi(x)+C|x|^\alpha-D$, with $\psi$ built from Fourier transforms of spheres, is used to prove extremality through a Plancherel rearrangement: whenever $H\ge 0$ and $H\,d\mu_*=0$, the candidate minimizes the $\alpha$-th moment among all competitors with the same norm and weighted Fourier bound. The explicit extremizers found by this method include the interval characteristic function for $(\alpha,1,1)$ with $\alpha\ge 2$, whose sharp constant is $1/((2\pi)^\alpha(\alpha+1))$, settling the interval-optimality conjecture from the problem's origin.

Load-bearing premise

The proof that the interval function is optimal rests on the claimed sign bound $1\ge c_k^{(\alpha)}(-1)^k(k+1/2)^3>0$ for its cosine coefficients, which the paper justifies by integration by parts and monotonicity but whose final sign is not fully established for every $k$.

Editorial extensions

If this is right

The sharp inequality holds for all parameters, so the optimal averaging function exists for every choice of $\alpha,\beta$ and every dimension, and the optimal constant is computable in principle as the minimum of an explicit convex problem.
In one dimension with $\beta=1$ and $\alpha\ge 2$, the interval characteristic function is the unique extremizer up to dilation and normalization; this confirms that uniform averaging over an interval is the smoothest average in the sense of the original problem.
Every radial extremizer either oscillates to the maximum of its weighted Fourier transform out to infinity or is a finite sum of spherical shell measures; this reduces the search for optimal averaging measures to two structured families.
The Fourier transform of every extremizer decays like $|\xi|^{-\beta}$ (with the exponent $\gamma$ replaced by $(d-1)/2$ when $\beta<(d-1)/2$), with nonzero limsup, answering the previously open decay question.
For the tent function in one dimension, the same certificate proves global extremality among positive-definite functions for $(\alpha,\beta)=(2,2)$, resolving a question left open in earlier work.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The certificate method is likely to transfer to discrete and semi-discrete variants of the averaging problem, since it only requires a positive trigonometric (or spherical-harmonic) certificate and a Plancherel identity; testing it on the discrete averaging problem with $\beta=2$ would be a direct next step.
The rigidity dichotomy suggests a computational strategy for finding extremizers: solve the dual convex program on a grid of spherical-shell atoms; if the answer converges to a sphere measure, Theorem 1.5(4) may hold beyond the stated ranges.
The interval-extremality conjecture could be made fully self-contained if the coefficient sign bound (3.5) were replaced by an exact evaluation of $c_k^{(\alpha)}$ in terms of hypergeometric or gamma functions; such a formula would also simplify the proof.
If the lower bound in Corollary 1.6 could be upgraded to a limit rather than a limsup, the decay statement would identify the exact order of the Fourier transform at infinity, which would be a new result beyond the paper.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

Here is my read on Saucedo–Tikhonov (arXiv:2508.17938). Short version: this is a good paper, the main results hold up, and the one point that might trip up a referee is a non-issue.

What is new: they prove C*_{α,β,d}>0 for all α,β>0 and existence of extremizers (Theorem 1.2), which was only known for β>d/2. They also give explicit extremizers in four regimes: the interval in d=1, α≥2, β=1 (confirming Steinerberger's conjecture), the tent function, the inverse-radius measure in R^3, and the sphere measure for certain parameters. Theorem 1.5 is a useful structural dichotomy: either the Fourier transform attains its weighted sup norm at arbitrarily large frequencies, or the extremizer is a finite sum of spherical shell measures. The duality framework in Section 4 is clean and gives the asymptotics as well.

The proof of Theorem 1.2 is exactly as solid as the reader's report says: tightness, Prokhorov, radial averaging. The explicit cases are handled by Lemma 3.2, which is a weak-duality certificate argument. The flagged issue is (3.5), the sign of the Fourier coefficients c_k. The reader worried that the paper just says '-g'''>0' and stops. On reading the actual integral, the step is fine: after three integrations by parts, the coefficient is (2πλ)^{-3} ∫ -g'''(x)[(-1)^k - sin(2πλx)] dx. For even k the bracket is 1 - sin θ ≥ 0; for odd k it is -1 - sin θ ≤ 0. Since -g'''>0, the product has the sign of (-1)^k, and it is not identically zero. So (3.5) holds and the certificate is non-negative. The proof could be expanded by half a sentence, but there's no mathematical gap.

One minor editorial flaw: the definition of c_k^{(α)} writes the integral from 0 to 1/2. As written, that gives half of the claimed value for α=2. It should be the integral over [-1/2,1/2] (or equivalently twice the given integral). The subsequent identities all use the correct convention, so it's just a typo, but a referee should flag it.

Bottom line: the paper deserves a serious referee. The main theorem is a genuine advance, the explicit extremizers are nontrivial, and the duality viewpoint is worth having. I would not hold the paper hostage to the (3.5) step. With the coefficient definition typo fixed, I'd expect it to go through.

Regards.

Referee Report

0 major / 5 minor

Summary. The paper proves a sharp Fourier uncertainty principle for finite non-negative Borel measures on R^d, with the quantity C_{α,β,d}(μ) = ||\hat{μ}(ξ)|ξ|^β||_∞^α I_μ(|x|^α)^β / ||μ||_{TV}^{α+β}. Theorem 1.2 establishes that the optimal constant C^*_{α,β,d} is positive for all α,β>0 and that a radial extremizer exists. Theorem 1.4 explicitly identifies extremizers in four parameter regimes, including the characteristic function of an interval in d=β=1 for α≥2, thereby confirming a conjecture of Steinerberger. Theorem 1.5 gives a structural dichotomy for extremizers: either the Fourier transform attains its norm at arbitrarily large frequencies or the extremizer is a finite sum of spherical-shell measures; Corollary 1.6 derives the resulting Fourier decay rates. The proofs use compactness via Prokhorov's theorem, a linear-programming weak-duality criterion for extremality, and detailed trigonometric inequalities.

Significance. If the results are correct, this is a substantial advance on a problem that has been open at the level of global extremizers. The explicit certificates in Lemma 3.2 are a particular strength: they reduce extremality to checking a non-negative comparison function, and they give closed-form extremizers and constants in several nontrivial regimes. The confirmation of Steinerberger's conjecture for α≥2 in the one-dimensional β=1 case is a clean, falsifiable statement. Theorem 1.5 and Corollary 1.6 provide new structural information about all extremizers, including a dichotomy between shell-type measures and measures whose Fourier transform reaches the maximal allowed growth at arbitrarily large frequencies. The paper is carefully written and the main line of argument is convincing.

minor comments (5)

[§3, proof of Theorem 1.4(1), inequality (3.5)] The proof of (3.5) is too terse: the sentence "(3.5) follows by noting that ... -g'''(x)>0" does not by itself explain why the oscillatory integral has the stated sign. The argument is valid, but it should be written out: after the third integration by parts one has c_k = (2πλ)^{-3} ∫_0^{1/2} (-g'''(x))[(-1)^k - sin(2πλx)] dx with λ=k+1/2, and the bracketed factor is nonnegative for even k and nonpositive for odd k, while -g'''(x)>0. The upper bound |c_k| ≲ (k+1/2)^{-3} also follows from |(-1)^k - sin(2πλx)|≤2. Please include these two or three lines.
[§6, Lemma 6.1 and Theorem 1.4(4)] The statement of Lemma 6.1 begins "Let β=1 and 2≥α≥α0", while Theorem 1.4(4) requires the union of the regime α∈[α0,2] with β=1 (handled by Lemma 6.1) and the regime α≥2, 0<β≤1 (handled by Lemma 6.3). The partition of the parameter ranges should be stated explicitly so the reader sees which lemma covers which part of the theorem.
[§4, Lemma 4.2, Step 4] In the proof that the zero set of H_0 has no accumulation point at t=0, if A-λ*≠0 the conclusion is immediate from continuity, and this case should be separated before considering the expansion with s∈{l,α,l+r}. In the remaining case A=λ*, the stated expansion does give the result, but the role of the constant term should be made explicit.
[Notation, Section 1] There is a typo in the definition of δ_{λ S^{d-1}}: "uniform uniform measure" should be "uniform measure".
[Throughout] The phrase "up to scaling and dilation" in Remark 1.7(i) is redundant; the intended meaning is "up to dilation and normalization", and the wording could be unified with the earlier uniqueness statement in Theorem 1.4(1).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified; the derivation is self-contained.

full rationale

The paper's central claims are self-contained. Theorem 1.2 derives the positivity of the optimal constant and the existence of extremizers via Prokhorov compactness directly from the definition of C* as an infimum, with no parameter fitted to force the value of C*. Theorem 1.4 uses Lemma 3.2, a weak-duality certificate: candidate extremizers and explicit nonnegative H functions are constructed, and the positivity of H is verified by direct trigonometric estimates in Section 6 and the Appendix. The Fourier coefficients c_k are fixed by the candidate function and are not chosen by fitting C*. Similarly, the duality argument in Section 4 proves the structural result Theorem 1.5 from the same variational formulation rather than importing a conclusion from the authors' own prior work. There are no load-bearing self-citations: the references to Steinerberger and Kravitz-Steinerberger are to prior work by other authors, and the one externally mentioned fact, assertion (3.5), is additionally proved inside the paper by an integration-by-parts estimate. No uniqueness theorem is imported from the authors' own papers, and no known result is merely renamed. I therefore find no step in which a prediction reduces by construction to its own input.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard harmonic analysis and convex duality: Prokhorov compactness, Fenchel-Rockafellar duality, Bessel function asymptotics, Ingham's inequality, and Bernoulli number estimates. No numbers are fitted to data. The radial symmetrization uses only averaging over the orthogonal group. No new entities are introduced.

assumptions (6)

standard math Prokhorov's theorem and tightness for probability measures on R^d (Billingsley [1]).
Used in Theorem 1.2 to extract a weakly convergent subsequence from a minimizing sequence.
standard math Fenchel-Rockafellar duality theorem (Villani [11]).
Used in Lemma 4.1 to obtain strong duality and slackness conditions for the extremization problem.
standard math Bessel function asymptotics, formula (4.13)-(4.14) (Grafakos [4]).
Used in Theorem 1.5 and Corollary 1.6 to analyze Fourier transforms of radial measures and their decay.
standard math Ingham's inequality for finite trigonometric sums (Jaming and Saba [5]).
Used in the proof of (4.15) to show the lim sup of a finite cosine sum is positive.
standard math Bernoulli number ratio estimate (Qi [8]).
Used in Lemma 6.2 to establish the x/tan x inequalities applied in Lemma 6.3.
domain assumption Propagation of regularity for positive definite functions (Massa, Peron, Piantella [7]).
Used in Proposition 5.2 to extend analyticity from a neighborhood of the origin to all of R^d.

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Cite this review

Pith. "Pith review of Extremizers of a Fourier uncertainty principle related to averaging." pith.science (2026). https://pith.science/paper/QTDLDIAH

@misc{pith2026250817938,
  author       = {Pith},
  title        = {Pith review of: Extremizers of a Fourier uncertainty principle related to averaging},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QTDLDIAH}},
  note         = {Machine review of arXiv:2508.17938}
}

abstract

We study the uncertainty principle $$\lVert\widehat{\mu}(\xi) |\xi|^\beta\rVert_\infty^{\alpha} \left(\int |x|^\alpha d \mu\right)^{\beta} \geq C(\alpha,\beta,d){\lVert{\mu}\rVert_{TV}^{\alpha+\beta}}$$ for finite non-negative measures on $\mathbb{R}^d $. We prove that $C(\alpha,\beta,d)>0$ for all $\alpha,\beta>0$ and that extremizers exist. Moreover, we obtain an abstract characterization of the extremizers, which allows us to describe their asymptotic behavior and, for certain parameter values, to determine them explicitly.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

11 extracted references · 9 canonical work pages

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P. Jaming and C. Saba. From Ingham to Nazarov’s inequality: a survey on some trigonometric inequalities. Adv. Pure Appl. Math. , 15(3):12–76, 2024

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[7]

Massa, A

E. Massa, A. P. Peron, and A. C. Piantella. Estimates on the derivatives and analyt- icity of positive definite functions on Rm. Anal. Math., 43(1):89–98, 2017

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2003

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Billingsley

P. Billingsley. Convergence of probability measures. Wiley Ser. Probab. Stat. Chich- ester: Wiley, 2nd ed. edition, 1999

work page 1999

[2] [2]

Boyd and L

S. Boyd and L. Vandenberghe. Convex optimization. Cambridge University Press, 2004

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Y.-K. Cho and Y. W. Park. The zeros of certain Fourier transforms: Improvements of P´ olya’s results. arXiv:2110.01885, 2021

work page Pith review arXiv 2021

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Grafakos

L. Grafakos. Classical and modern Fourier analysis . Upper Saddle River, NJ : Pear- son/Prentice Hall, 2004

work page 2004

[5] [5]

Jaming and C

P. Jaming and C. Saba. From Ingham to Nazarov’s inequality: a survey on some trigonometric inequalities. Adv. Pure Appl. Math. , 15(3):12–76, 2024

work page 2024

[6] [6]

Kravitz and S

N. Kravitz and S. Steinerberger. The smoothest average: Dirichlet, Fej´ er and Cheby- shev. Bull. Lond. Math. Soc. , 53(6):1801–1815, 2021

work page 2021

[7] [7]

Massa, A

E. Massa, A. P. Peron, and A. C. Piantella. Estimates on the derivatives and analyt- icity of positive definite functions on Rm. Anal. Math., 43(1):89–98, 2017

work page 2017

[8] [8]

F. Qi. A double inequality for the ratio of two non-zero neighbouring Bernoulli num- bers. J. Comput. Appl. Math. , 351:1–5, 2019. 30 MIQUEL SAUCEDO AND SERGEY TIKHONOV

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