Optimal concentration in the Paley-Wiener space

Lu\'is Daniel Abreu; Michael Speckbacher

REVIEW 1 major objections 4 minor 1 cited by

Optimal concentration in the Paley-Wiener space

T0 review · 1 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Intervals maximize the L²-concentration of band-limited functions on the real line, for any measurable set of fixed measure.

desk verdict If the proof holds, this settles the Donoho–Stark interval conjecture in full generality; the circle theorem is new and the variational core is sound — but the printed D_N definition has a real typo and Lemma 3.2 deserves an independent check. read the letter →

arxiv 2607.19192 v1 pith:T4RPUPAK submitted 2026-07-21 math.CA math.FA

Luís Daniel Abreu , Michael Speckbacher This is my paper · ORCID

classification math.CAmath.FA MSC 42A3842C0546E22

keywords Paley-Wienerspaceband-limitedfunctionsconcentrationinequalitiesoptimalintervalsanalytictrigonometricpolynomialsreproducingkerneluniversalitylimituncertaintyprinciples

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

The paper proves that among all measurable sets E of a given finite measure, a plain interval I of the same measure captures the largest possible fraction of the L² mass of any band-limited function. This settles a conjecture from 1989 that previously had been proved only under small time-bandwidth restrictions. The proof works by first showing the analogue for analytic trigonometric polynomials on the circle — intervals are optimal there too — and then transferring the result to the real line via a universality-type limit of the polynomial reproducing kernel to the sinc kernel. If correct, it gives a complete answer to the classic concentration problem in Paley–Wiener space and sharpens quantitative uncertainty principles.

What carries the argument

The universality-type identity lim_{N→∞} (1/(2N+1)) K_N(e^{iπt/N}, e^{iπξ/N}) = sin π(t−ξ)/(π(t−ξ)) for the reproducing kernel K_N(z,w)=Σ_{j=−N}^{N} (z w)^j of analytic trigonometric polynomials. This identity is used twice: it motivates the circle theorem and it supplies a controlled limit (midpoint Riemann sums of the sinc kernel) that pushes the circle result to the real line. The circle theorem itself rests on a zero-saturation lemma forcing an extremal density to consume all N available zeros, the one-hill lemma describing lower level sets, and an exact negative second-variation computation for relative translations of root blocks.

What would settle it

For some N≥1, take a measurable E⊂T of measure 0<µ<2π, compute the largest eigenvalue of the concentration operator for analytic polynomials of degree N on E, and compare with the eigenvalue for the circle interval of length µ. A single non-interval E with larger eigenvalue would falsify the circle theorem and, through the limit, the main theorem. A numerical search for small N and µ could look for such a counterexample.

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Extended reading notes

Core claim

On the circle, for analytic trigonometric polynomials of degree at most N, the paper establishes that the largest eigenvalue of the concentration operator is maximized by a circle interval (Theorem 2.1). The key structural fact is that an extremal density must be the square of a real amplitude with N zeros, so its lower-level set has exactly one component; any configuration with two or more root blocks can be perturbed by relative translations, yielding a negative second variation that contradicts local minimality. The real-line statement then follows by expanding circles: the Paley–Wiener sinc kernel is a uniform limit of the midpoint Riemann sums of the polynomial kernels, and the operator

Load-bearing premise

The entire proof depends on the claim that a global minimizer of the lower-tail functional must use every one of its N zero degrees of freedom — if a minimizer could leave even one degree unused, the subsequent one-component and negative-second-variation argument would not go through.

Editorial extensions

If this is right

Confirms the 1989 interval-extremal conjecture in full generality; previous restrictions on the time-bandwidth product are no longer needed.
Gives a sharp, computable bound: the best concentration on a set E is the concentration on the interval of the same measure, so interval constants are the universal benchmark.
Provides an independent optimal-concentration theorem for analytic trigonometric polynomials, useful as a finite-dimensional sharp form.
Establishes the universality-limit transfer as a proof technique for carrying extremal results from the circle to the real line in Paley–Wiener spaces.

Reading between the lines

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

The same circle-to-line transfer might apply to other reproducing kernel Hilbert spaces whose kernels admit universality limits, yielding interval-type optimizers in those spaces — the paper does not claim this.
The quantitative negative second variation suggests a stability statement: sets that nearly attain the maximal concentration must be close to intervals, but the paper does not derive such a bound.
For large degree N and later large circle radius, the construction may be viewed as an expanding-circle discretization of the classical prolate spheroidal problem; one could test numerically whether the extremal densities converge to the prolate spheroidal wave functions on intervals.
The proof's reliance on factorizing nonnegative trigonometric polynomials into squared analytic-modulus factors suggests that analogous concentration results for weighted polynomial spaces could be approached by the same zero-saturation framework.

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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

Short version: this is a serious paper, possibly a major one. The main theorem removes every restriction on |Ω||E| in the Donoho–Stark concentration problem. I read the proof carefully, and the structure is genuinely new: a sharp concentration theorem for analytic trigonometric polynomials on the circle, then a controlled expanding-circle limit that transfers the result to the line. The transfer is clean; the Riemann-sum estimate and the Hilbert–Schmidt bound are routine and correct. The circle theorem itself is a real result, not a stepping stone you could have gotten from prior work.

The variational engine is the most interesting part. The zero-saturation lemma (Lemma 3.2) is the load-bearing wall. I checked the two-sided perturbation argument, the concavity, the maximal-zero selection, and the contradiction. I could not find a gap. The one-hill lemma, the endpoint-velocity formulas, and the Hessian with the boundary covariance all check out. The algebraic identity in Section 8 is exact; the translation-invariant direction is handled correctly. There are no fitted constants, no hidden parameters, and the existing partial results are cited as benchmarks, not used as inputs. That is real evidence of honesty.

Soft spots. First, a genuine textual error: the definition of D_N in §2.2 prints c_{−k} = c_k. That makes eq. (9) false, as the counterexample |1 + i e^{it}|^2 shows. The intended condition is c_{−k} = \overline{c_k}. This is fixable, but it must be fixed; as printed, a reader cannot verify the claimed equivalence with normalized energy densities. Second, Lemma 3.2 is intricate enough that I would want an independent referee to walk through it line by line. The failure mode would be a subtle sign error in the perturbation or an overlooked case in the touching argument. I didn't find one, but the lemma is doing a lot of work. Third, the AI disclosure is unusually transparent, and the retained "graphic language" remnant should be edited out. That is a presentation issue, not a mathematical one, but it does raise the prior on subtle errors — which makes the needed referee scrutiny more important, not less.

Who is this for? Harmonic analysts, people working on spectral concentration, uncertainty principles, and shape optimization for localization operators. If the proof is correct, it closes a conjecture from 1989 that has resisted the rearrangement and isoperimetric methods. The paper deserves a serious referee. I would send it to a strong journal and specifically ask referees to attack Lemma 3.2 and the D_N definition. With the typo fixed and the AI remnants cleaned, I would be comfortable citing it.

Referee Report

1 major / 4 minor

Summary. The paper proves that, among all measurable sets of fixed finite measure, intervals maximize the L^2-concentration of band-limited functions in the Paley–Wiener space PW(Ω). This is Donoho–Stark's 1989 conjecture, previously known only under restrictions on the time–bandwidth product |Ω||E|. The proof proceeds in two stages: first a circle-level theorem (Theorem 2.1) asserting that intervals maximize concentration for analytic trigonometric polynomials of degree N; second a universality-type limit that transfers the circle result to the real line by approximating the Paley–Wiener sinc kernel by midpoint Riemann sums of polynomial kernels on expanding circles. The circle theorem is obtained through a variational problem for the lower-tail functional L_μ on normalized nonnegative trigonometric polynomials, an intricate zero-saturation lemma (Lemma 3.2) showing that an extremal density must have the maximal number of roots, and a negative second-variation argument for relative translations of root blocks.

Significance. If correct, the result resolves a 35-year-old conjecture in full generality and removes the time–bandwidth restriction that has persisted since Donoho–Stark. The circle theorem is also a substantial result of independent interest. A notable strength of the paper is that the proof is self-contained and parameter-free: the only external inputs are classical (Fejér–Riesz factorization, the bathtub principle, the implicit function theorem), and the final contradiction is an explicit negative sum of squares. The rigorous kernel approximation in Section 9 is carefully quantified and does not rely on the heuristic universality formula quoted in the introduction. The manuscript is unusually transparent about the provenance of its ideas, including the role of AI-assisted exploration, which does not affect the mathematical content.

major comments (1)

[§2.2, definition of D_N and eq. (9)] The set D_N is defined with the condition c_{-k}=c_k. For a real-valued nonnegative trigonometric polynomial, however, the Fourier coefficients satisfy c_{-k}=\overline{c_k}; the stronger condition c_{-k}=c_k excludes legitimate densities such as |1+i e^{it}|^2, whose coefficients are c_1=i, c_{-1}=-i. As printed, eq. (9), Λ_N(μ)=min_{g∈D_N} L_μ(g), is therefore false because the minimum is taken over a proper subset of the normalized densities obtained from Fejér–Riesz factorization. This is a load-bearing definition: Lemma 2.4, Lemma 3.2, and the identification of D_N with normalized energy densities all rely on it. The fix is local and does not affect the structure of the proof: replace c_{-k}=c_k by c_{-k}=\overline{c_k} (equivalently, state explicitly that g is real-valued). With this correction, the compactness, concavity, and zero-saturation arguments remain valid. The authors sho

minor comments (4)

[§9, eq. (49)] The symbol M_n in Δ_n = 1/M_n is undefined; from the subsequent midpoint Riemann sum construction one needs Δ_n = 1/(2n+1) = 1/L_n. Please define M_n or replace it by L_n.
[Introduction, universality display] The displayed limit 1/(2N+1) K_N(e^{iπt/N}, e^{iπξ/N}) = sinπ(t−ξ)/π(t−ξ) is not correct as written, since with w=e^{iπξ/N} the exponent contains t+ξ. The intended statement requires the conjugate variable, e.g. K_N(e^{iπt/N}, e^{-iπξ/N}), or an equivalent convention. The rigorous Step 2 of Section 9 uses a correctly defined midpoint Riemann sum, so this is not a gap in the proof, but the motivational formula should be corrected.
[§3.1–3.2] The phrase “real trigonometric polynomial” is used ambiguously: it sometimes means real-valued and sometimes seems to imply real Fourier coefficients. Given the correction to D_N, the authors should consistently say “real-valued” (or “Hermitian-symmetric coefficients”) to avoid confusion.
[§1, final paragraph] The sentence about the preliminary version “still keeps part of the interesting graphic language used by the Language Model” is informal and should be removed or rewritten in standard mathematical prose.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the circle theorem is derived from a variational problem solved in-line, and the Paley–Wiener result is a controlled limit proven directly in §9; self-citations are contextual only.

full rationale

The derivation chain is self-contained. Theorem 1.1 is obtained from Theorem 2.1 by the controlled limit in §9: the projection kernel k_n is shown to be the midpoint Riemann sum for the sinc kernel and to converge uniformly on bounded difference sets (eq. (52)), which is upgraded to operator-norm convergence via the Hilbert–Schmidt estimate (54). The limiting kernel identity (1) is classical but is re-proven here in the Riemann-sum form; the Levin–Lubinsky citation is flagged only as context ('for limits of this nature') and is not an input to the proof. Theorem 2.1 is proven from the definition of the concentration constant: complementation (6) and the variational equivalence (7)–(9) are equalities by construction, not assumptions. The extremal density is then found by an in-line chain — bathtub principle (Lemma 2.3, proven), compactness/concavity (Lemma 2.4, proven), zero saturation (Lemma 3.2), one-hill lemma (Lemma 4.1), block-translation Hessian (46) — using only Fejér–Riesz (classical, cited to an external survey [8]), the implicit function theorem, and real-analyticity. No parameter is fitted to the target result and no 'prediction' is read back from a fit. The only self-citations ([1], [2], [4]) appear in the introduction as historical context for related Fock/Bergman/Hardy problems; none enters the proof as an assumption, so they are not load-bearing. A textual defect is present: D_N is printed with 'c_{-k}=c_k', whereas the Hermitian condition c_{-k}=\overline{c_k} is needed for eq. (9) to hold for all densities |p|^2; this is a correctness/typographical issue in the written manuscript, not a circularity — the variational and perturbation argument itself does not presuppose its conclusion. No circular step was found; score 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof's inputs are classical theorems and standard analysis: Fejér–Riesz factorization (§2.2), the analytic implicit function theorem (§6), the sinc reproducing kernel (eq. (1)), spectral equivalences (eq. (3)), and real-analytic level-set continuity (§4.2). No free parameters are fitted and no new entities are introduced: ρ in §4.2 is determined by the constraint |F*| = μ, and the endpoint branches e(c,τ) and ρ(c) in §6 are determined by the implicit-function theorem and the measure constraint. Caveat: the printed definition of D_N (c_{−k} = c_k) makes the Fejér–Riesz identification (9) false; with the evident intended condition c_{−k} = \overline{c_k}, the ledger is as listed.

assumptions (5)

standard math Fejér–Riesz factorization: every nonnegative trigonometric polynomial of degree at most N is |p(e^{it})|² for some p ∈ A_N (Theorem 2.2, cited to Dritschel–Rovnyak [8]).
Invoked at §2.2 to pass from the operator extremal problem (7) to the density lower-tail problem (8)–(9). Note: as printed D_N requires c_{−k} = c_k; the factorization statement only reconciles with (9) under the intended condition c_{−k} = \overline{c_k}.
standard math Analytic implicit function theorem: endpoint branches e(c,τ) and the level function ρ(c) with Φ(c,ρ(c)) = μ exist analytically near (0,ρ) (§6, eq. (24)).
Load-bearing in §6–§7 for the translation family q_c and the Hessian computation; requires transverse crossings (17), i.e., q′(e) ≠ 0 at the level.
standard math Spectral equivalences: ‖M_E P_Ω‖² = ‖M_E P_Ω M_E‖ = ‖P_Ω M_E P_Ω‖, and equality of nonzero spectra with the integral operator on L²(E) with kernel k(x−y) (eqs. (3), (48)).
Used to switch between operator norms on different spaces in §9, in particular (53) Q_n*Q_n vs Q_nQ_n* and (56) the truncation estimate.
standard math The sinc kernel k(x−y) = ∫_{−1/2}^{1/2} e^{2πiξ(x−y)}dξ is the reproducing kernel of PW([−1/2,1/2]) (eq. (1)).
Domain assumption: definition of the Paley–Wiener space and its projection operator.
standard math Level-set measure continuity: for nonconstant real-analytic q, |{q² = γ}| = 0 for every γ, so ρ with |{q² ≤ ρ²}| = μ exists (§4.2).
Needed to construct the bathtub minimizing set F* = {q² ≤ ρ²} with exact measure μ; standard real analysis.

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Cite this review

Pith. "Pith review of Optimal concentration in the Paley-Wiener space." pith.science (2026). https://pith.science/paper/T4RPUPAK

@misc{pith2026260719192,
  author       = {Pith},
  title        = {Pith review of: Optimal concentration in the Paley-Wiener space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T4RPUPAK}},
  note         = {Machine review of arXiv:2607.19192}
}

abstract

Let $\Omega \subset \mathbb{R}$ be a bounded interval and let $PW(\Omega )$ be the corresponding Paley--Wiener space. For a measurable set $E\subset \mathbb{R}$ of finite measure, consider the largest possible fraction of the $L^{2}$-mass of a function in $PW(\Omega )$ that can lie in $E$. We prove that this concentration is no larger than the concentration attained on an interval of measure $\lvert E\rvert $. Thus, \emph{intervals optimize concentration in the Paley-Wiener space of band-limited functions.} The proof, based on an universality-type limit of the reproducing kernel of analytic trigonometric polynomials on the circle, has two steps. First, we establish an \emph{optimal concentration theorem for analytic trigonometric polynomials on the circle}. Second, the universality-type limit transfers the result from the circle to the real line, by controlling the expansion of circles whose projection kernels are midpoint Riemann sums for the Paley--Wiener sinc kernel.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

The isoperimetric inequality for the Ky Fan norm
math.FA 2026-07 accept novelty 7.0 of 10

Among measurable sets of fixed area, the disc uniquely maximizes (up to translation) the Ky Fan norm of the Fock Toeplitz operator, confirming the Nicola–Riccardi–Tilli conjecture.

Reference graph

Works this paper leans on

17 extracted references · 1 linked inside Pith · cited by 1 Pith paper

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

L. D. Abreu, and M. Speckbacher,Donoho-Logan large sieve principles for modulation and polyanalytic Fock spaces, Bull. Sci. Math. 171, 103032, (2021)

2021

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2012

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Baeza–Guasch,On an uncertainty result by Donoho and Stark, arXiv:2307.04558, (2023)

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arXiv 2023

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1988

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Daubechies, and T

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[6] [6]

D. L. Donoho, and P. B. Stark,Uncertainty principles and signal recovery, SIAM J. Appl. Math. 49(3), 906–931, (1989)

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D. L. Donoho, and P. B. Stark,A note on rearrangements, spectral concentration, and the zero-order prolate spheroidal wavefunction, IEEE Trans. Inform. Theory 39(1), 257–260, (1993)

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2010