The H\"ormander--Bernhardsson function in higher dimensions

Antonio Pedro Ramos; Danylo Radchenko; Felipe Gon\c{c}alves

REVIEW 3 major objections 3 minor 1 cited by

The H\"ormander--Bernhardsson function in higher dimensions

T0 review · 3 major / 3 minor · reviewed 2026-08-27 · deepseek-v4-flash

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Pith's one-line read For every dimension d≥1, the radial extremal function of the point-evaluation problem in Paley–Wiener space satisfies an explicit third-order linear ODE with polynomial coefficients.

desk verdict Gonçalves–Radchenko–Ramos closes the even-dimensional gap with a genuinely different proof; the announced ODE holds, but Proposition 3's PW membership needs a written proof. read the letter →

arxiv 2608.22198 v1 pith:NU4ZZNYB submitted 2026-08-23 math.CA math.CVmath.FA

Felipe Gonçalves , Danylo Radchenko , Antonio Pedro Ramos This is my paper · ORCID

classification math.CAmath.CVmath.FA MSC 30D1533E3034A3041A4442A05

keywords Hörmander–BernhardssonfunctionPaley–Wienerspacepointevaluationextremalconstantthird-orderODEinterpolationformulazerosequenceweighted

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 studies the sharp constant $C_d$ for point evaluation in the Paley–Wiener space $PW^1(\mathbb{R}^d)$ — the space of entire functions of exponential type 1 integrable on $\mathbb{R}^d$ — and proves that the unique radial extremizer $\varphi_d$ satisfies a third-order linear ODE with polynomial coefficients for every dimension $d\geq 1$. The ODE extends a recent odd-dimensional result to even dimensions, where the earlier method (which relied on $|x|^{d-1}$ being a polynomial) does not apply. The route is new: an interpolation formula samples admissible functions at the zeros of $\varphi_d$, reciprocal expansions factor $\varphi_d$ into two entire functions, and the ODE emerges as the symmetric square of second-order equations for the factors. The result gives the first unified analytic description of the extremal function across all dimensions and supplies a concrete equation from which finer properties of $C_d$ can be studied.

What carries the argument

The machinery is the interpolation formula of Proposition 4, which states that for every $f\in PW_{d-1}$, $$\frac{f(z)}{\varphi_d(z)}=f(0)+f'(0)z+\frac{$z^{2}$}{2c_d}\sum_{n\geq 1}(-1)^n $t_n^{{d-1}}$\left[\frac{f(t_n)}{z-t_n}-\frac{f(-t_n)}{z+t_n}\right],\qquad c_d=\frac{d}{4C_d}.$$ This formula encodes the Euler–Lagrange equation as a sampling series of interpolation type over the zero sequence $\{t_{d,n}\}$. Applying it to carefully truncated test functions modelled on $z^{-d-1}$ yields the reciprocal expansions of Propositions 6 and 7, which in turn motivate the dimension-dependent factorization $\varphi_d=\Phi_+\Phi_-$ into two entire functions of exponential type 1. Lemma 8 gives each factor a second-order ODE with an entire error term; growth estimates for entire functions of exponential type force that error to be a polynomial of degree at most $d+1$, and the asymptotic zero-counting law for the extremal function fixes its leading coefficient, so the symmetric square of the second-order equation is exactly the ODE of Theorem 1.

What would settle it

Decisive check: for $d=2$, compute $\varphi_2$ and its zeros numerically, take the even test function whose Fourier transform is $(1-\xi^2)^4$ on $[-1,1]$ (so it lies in $PW_1$ with strong decay), and compare $f(z)/\varphi_2(z)$ with the right-hand side of Proposition 4 at several non-real $z$. A mismatch beyond numerical round-off would refute the interpolation formula and with it the proof of Theorem 1; an accurate match would support the theorem.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1: for every $d\geq 1$, the Hörmander–Bernhardsson function $\varphi_d$ — the unique even, real-entire, exponential-type-1 minimizer of the weighted one-dimensional problem (1), normalized by $\varphi_d(0)=1$ — satisfies $$$z^{{2d+2}}$\varphi_d'''(z)+3(d+1)$z^{{2d+1}}$\varphi_d''(z)+\bigl((d+1)(2d+1)$z^{{2d}}$-r(z)\bigr)\varphi_d'(z)-\tfrac12 r'(z)\varphi_d(z)=0,$$ where $r(z)=-z^{2d+2}+\frac{d^2}{4C_d^2}\sum_{j=0}^d \alpha_j z^{2j}$ and $\alpha_j$ is the coefficient of $z^{2j}$ in $1/\varphi_d(z)^2$. Equivalently, $\varphi_d$ satisfies the two first-order identities displayed in the theorem. The proof derives an interpolation formula (Proposition 4) from the Euler–Lagrange equation, uses it to obtain reciprocal expansions that suggest a factorization $\varphi_d=\Phi_+\Phi_-$, proves a second-order ODE for each factor, and recovers the third-order equation as the symmetric square. This is the first characterization of the extremizer that holds uniformly in even and odd dimensions.

Load-bearing premise

The load-bearing premise is that the auxiliary function $g_z(w)=\frac{z f(z)\varphi_0(w)-w f(w)\varphi_0(z)}{w^2-z^2}$ belongs to the admissible space $PW_{d-1}$ for every $z$; the paper asserts this without proof, and the interpolation formula built on it drives the reciprocal expansions and the entire ODE derivation.

Editorial extensions

If this is right

In dimension $d=1$, Theorem 1 recovers the known third-order equation for the Hörmander–Bernhardsson function obtained from the factorization $\varphi_1(z)=\Phi(z)\Phi(-z)$, so the new statement contains the one-dimensional case.
For odd $d$, the reciprocal formula in Proposition 6 is equivalent to the quadratic functional identity of [9, Theorem 1(a)], so the new derivation is consistent with the existing odd-dimensional characterization while also covering even dimensions.
Proposition 4 shows that every admissible function in $PW_{d-1}$ is determined by its values at the zeros $t_{d,n}$ through a sampling series of interpolation type, so the zero sequence gives complete sampling information in the weighted space for all $d$.
The factorization $\varphi_d=\Phi_+\Phi_-$ holds in every dimension, with the factors built from alternating or parity-split subsequences of the zeros, and each factor has exponential type 1.

Reading between the lines

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

Since Remark 5 states that every result through the interpolation formulas holds for real $d>0$, the same proof scheme may yield a continuous family of ODEs for weights $|x|^s$ with non-integer $s$, unifying the even/odd split into one analytic family.
A numerical test of the ODE of Theorem 1 for $d=2$ or $d=4$ could determine whether the equation, together with $\varphi_d(0)=1$ and the known zero spacing, uniquely determines the extremizer; if it does, the ODE becomes a practical tool for computing $C_d$ to high precision and for conjecturing its asymptotic rate.
The reciprocal formulas for even $d$ split the zeros into two alternating subsequences; a natural next step is to derive discrete equations for these subsequences, which may expose finer spacing statistics of the extremal zeros than the infinite product factorization does.

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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

The main thing you should know: this paper proves the Hörmander–Bernhardsson ODE for the radial extremizer of the point-evaluation constant in every dimension, closing the even-dimensional case that Gorbachev's method couldn't reach. The route is genuinely different—interpolation and reciprocal formulas instead of a spectral-gap argument—and the odd-dimensional overlap with Gorbachev's preprint is disclosed plainly. The proof is a derivation from the Euler–Lagrange equation, not a fitted ansatz: the polynomial r(z) is recovered from the extremizer's Taylor coefficients, so there is no circularity.

What's good. The main theorem is new for even d and unifies the characterization. The interpolation formula (Prop 3) is the engine, and the factorization φ = Φ_+ Φ_- with different zero-splittings for even/odd parity is a nice idea. The coefficient comparison in Lemma 10 and the zero-counting limit are serious, honest calculations. The appendix distributional identity gives an independent route to Proposition 2, which is a plus. Self-citations are appropriate—[4,7] supply background that is largely reproved here.

Soft spots. Proposition 3 asserts without proof that g_z(w) = (z f(z) φ_0(w) − w f(w) φ_0(z))/(w^2 − z^2) lies in PW_{d-1}. The stress-test note worries this is load-bearing. On reading the paper, the membership is true: h(w)=f(z)φ(w)−f(w)φ(z) is a linear combination of two PW_{d-1} functions, hence in PW_{d-1}; it has zeros at ±z, so division by (w^2−z^2) preserves the Fourier support (standard division-by-linear-factor for Paley–Wiener functions), and multiplication by w just differentiates the Fourier transform, preserving support. Weighted L^1 integrability follows from the improved 1/x^2 decay. But the paper should state and prove this as a lemma—the one-line '∈ PW_{d-1}' is too fast for a step that everything downstream rests on. Similarly, existence of extremizers is dismissed as 'a standard argument' in Section 2.1; worth a sentence or a reference to [7]. Neither issue is fatal—I checked the logic of the chain and it holds up—but both should be fixed before publication. Lemma 9's Phragmén–Lindelöf argument is compressed; a referee should ask for the constants to be tracked more carefully, though the estimates look right.

Bottom line: this is a correct and significant completion of Gorbachev's result. The even-dimensional ODE is new, the method is more natural than the spectral-gap approach, and the paper is honest about what overlaps with [9]. It deserves a serious referee and, after a minor revision adding the missing lemma and existence justification, publication.

Who it's for: harmonic analysts working on Paley–Wiener spaces, extremal problems, and entire functions of exponential type. I'd cite it if I were working on point-evaluation constants.

My recommendation: send it to peer review. A good referee will catch the missing lemma but won't find a hole in the main argument.

Referee Report

3 major / 3 minor

Summary. The paper studies the norm of the point evaluation functional on the Paley--Wiener space PW^1(R^d), reduces the problem to a weighted one-dimensional problem, and studies the unique extremizer φ_d, called the Hörmander–Bernhardsson function. The main result, Theorem 1, states that φ_d satisfies a third-order linear ODE with polynomial coefficients for every d ≥ 1, extending Gorbachev's odd-dimensional result. The proof proceeds by deriving a summation formula (Proposition 2), an interpolation formula (Propositions 3–4), reciprocal formulas (Propositions 6–7), a factorization φ_d = Φ_+ Φ_-, and finally the ODE by a symmetric-square argument. Lemma 10 identifies the polynomial r(z) by a power-series comparison and a zero-counting asymptotic.

Significance. If the gaps identified below are repaired, Theorem 1 is a substantial advance: it supplies the first unified differential-equation characterization of the extremal function in both even and odd dimensions, and the interpolation and reciprocal formulas are likely to be reusable tools. The proof is genuinely variational rather than tautological: the ODE is derived from the Euler–Lagrange equation, and the coefficients of r(z) are identified from the extremizer itself. The paper is also commendable for treating the parity issue explicitly and for attempting to be self-contained.

major comments (3)

[Section 3.2, Proposition 3] The assertion that the divided-difference function g_z(w) = (z f(z) φ_0(w) − w f(w) φ_0(z))/(w^2 − z^2) lies in PW_{d−1} is stated without proof. This is not a formal consequence of f, φ ∈ PW_{d−1}: at infinity g_z(w) behaves like (f(z)φ(w) − f(w)φ(z))/w, and the stated hypotheses do not by themselves guarantee integrability against |w|^{d−1} dw. Since Proposition 2 is an identity valid only for functions in PW_{d−1}, and Proposition 3 feeds directly into the reciprocal formulas, Lemma 8, and the ODE of Theorem 1, this missing membership estimate is load-bearing. Please add an explicit lemma with a quantitative decay estimate, or provide an approximation argument that justifies applying the summation formula to g_z.
[Section 3.2, Eq. (5), and Section 3.1] The derivation of equation (5) applies Proposition 2 to f_n(z) = φ_0(z)/(z^2 − t_n^2). The function f_n is entire of exponential type, but on the real line it behaves like φ(x)/x; membership in PW_{d−1} is not established and is not implied by φ ∈ PW_{d−1}. For d = 1 this would require ∫ |φ(x)|/|x| dx < ∞, which is stronger than φ ∈ L^1. The same unproved division by a linear factor occurs in the derivation of Proposition 2 itself, where identity (4) is applied to f(x)/x. These steps are not merely technical: without a division lemma or a limiting argument, the summation formula and all later interpolation identities rest on an unverified premise.
[Section 2.1] The paper states that 'a standard argument shows that extremizers of (3) must exist' but gives neither the argument nor a specific reference. The existence of a minimizer is a prerequisite for defining φ_d and hence for the uniqueness and ODE arguments. If existence is already proved in [7] or [9], please cite the precise statement; otherwise include the compactness argument (for example, via the Paley–Wiener description and weak-* compactness of the relevant unit ball).

minor comments (3)

[Section 5] The appendix proves Proposition 2 via a distributional identity and then says a 'routine approximation argument' completes the extension to PW_{d−1}; please spell out the approximation, since the interpolation formulas in Section 3 are applied to functions that are not compactly supported in frequency and need the full summability statement.
[Section 4, Lemma 10] In the proof of Lemma 10 the notation φ1 is used for the derivative without a local definition; please write φ' consistently.
[General] The title and abstract contain several broken spacing artifacts ('H ¨ormander'), and the paper would benefit from a final proofreading pass to normalize such typographical issues.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity: the ODE is derived from the variational Euler-Lagrange system, and the main proof dependencies do not reduce to self-citations or fitted inputs; the unproved g_z membership in Section 3.2 is a correctness gap, not a circular one.

full rationale

No circular step is present. Theorem 1 is derived from the Euler-Lagrange/first-variation identity (4) through Proposition 2, the interpolation formula (Proposition 3), the reciprocal formulas (Propositions 6 and 7), and the symmetric-square argument in Section 4. The coefficients alpha_j and the constant C_d are defined from the extremizer and the extremal constant, not fitted to force the ODE; Lemma 10 identifies r(z) by coefficient comparison inside the already-proved differential identity, which is a characterization rather than a tautology. The only self-citations to [4] and [7] are contextual, and the cited properties are either reproved in Section 2 or used only as background, so no load-bearing self-citation occurs. The manuscript does contain a genuine technical gap at Section 3.2, where the divided-difference function g_z(w) = (z f(z) phi_0(w) - w f(w) phi_0(z))/(w^2 - z^2) is asserted to lie in PW_{d-1} without proof, and Section 2.1 delegates existence of extremizers to 'a standard argument'. These are correctness risks that could invalidate the proof if false, but they are not cases of a prediction reducing to its inputs, so they do not affect the circularity score.

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

The central claim rests on standard entire-function theory (Krein factorization, Lindelöf, Cartwright zero counting) and on two domain assumptions that the paper states but does not fully prove: existence of extremizers, and the Paley-Wiener membership of the divided-difference functions used in interpolation. No free parameters are fitted; C_d and α_j are defined from the extremal problem, not tuned.

assumptions (5)

domain assumption Existence of an extremizer for the weighted L1 variational problem (1)
Invoked in Section 2.1 with 'A standard argument shows that extremizers of (3) must exist'; defines the object φ_d.
domain assumption First-variation identity (4) applies to derivatives and divided differences of functions in PW_{d-1}
Used in Proposition 2 with f' and x^{-1}f, and in Proposition 3 with g_z; the required weighted integrability is asserted rather than proved.
standard math Krein's factorization theorem: a real entire function of exponential type that is nonnegative on R factorizes as h(z)h(conj z) with h of half the type
Used in Section 2.2 to prove uniqueness of the extremizer from sign agreement.
standard math A ratio of two entire functions of exponential type, when entire, has exponential type (Lindelöf)
Used in Lemma 8, cited from [11, p. 22], to conclude the auxiliary function q is of exponential type.
standard math For a type-1 entire function bounded on R with only real zeros, the zero-counting function satisfies n(r)=r/π+o(r)
Used in Lemma 10 via [12, p. 127] to compute the leading coefficient β_{d+1} of r.

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Cite this review

Pith. "Pith review of The H\"ormander--Bernhardsson function in higher dimensions." pith.science (2026). https://pith.science/paper/NU4ZZNYB

@misc{pith2026260822198,
  author       = {Pith},
  title        = {Pith review of: The H\"ormander--Bernhardsson function in higher dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NU4ZZNYB}},
  note         = {Machine review of arXiv:2608.22198}
}

abstract

We study the problem of finding the norm of the point evaluation operator in the Paley--Wiener space $PW^{1}(\r^d)$, consisting of $d$-variable functions of spherical exponential type that are integrable on $\r^d$. The extremal functions can be taken radial, which naturally leads us to consider a related extremal problem in a weighted Paley--Wiener space of single-variable functions. We establish that the radial extremal function must satisfy a third-order linear ODE with polynomial coefficients for every $d \geq 1$, extending Gorbachev's recent odd-dimensional result. Along the way, we prove interpolation and reciprocal formulas involving the zeros of the extremizer.

Discussion (0). Continue with ORCID to comment.

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

Exact Asymptotics of the Multidimensional Nikolskii Constant
math.CA 2026-09 accept novelty 8.0 of 10

As d tends to infinity, 2^d times the normalized L1 Nikolskii constant converges to π/2, and the rescaled zeros of the extremal function converge to an explicit Bessel-type measure.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.

[1] [7]

F. Dai, D. Gorbachev, and S. Tikhonov, Estimates of the asymptotic Nikolskii constants for spher- ical polynomials, J. Complexity65(2021), 101553

work page 2021

[2] [9]

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work page Pith review arXiv 2026

[3] [1]

N. N. Andreev, S. V. Konyagin, and A. Yu. Popov, Extremal problems for functions with small support, Math. Notes60(1996), no. 3, 241–247

work page 1996

[4] [2]

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A. Bergman, H¨ ormander’s inequality and point evaluations in de Branges space, Rev. Mat. Iberoam. 42(2026), no. 2, 551–572

work page 2026

[5] [3]

Bondarenko, J

A. Bondarenko, J. Ortega-Cerd` a, D. Radchenko, and K. Seip, The H¨ ormander–Bernhardsson ex- tremal function: a preliminary study, inRecent advances in approximation and potential theory, pp. 77–88, Birkh¨ auser, Cham, 2026

work page 2026

[6] [4]

(2026), To appear

A. Bondarenko, J. Ortega-Cerd` a, D. Radchenko and K. Seip, The H¨ ormander–Bernhardsson ex- tremal function, preprint at arXiv:2504.05205

work page arXiv

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O. F. Brevig, A. Chirre, J. Ortega-Cerd` a, and K. Seip, Point evaluation in Paley–Wiener spaces, J. Anal. Math.153(2024), no. 2, 595–670

work page 2024

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Carneiro, M

E. Carneiro, M. B. Milinovich, and K. Soundararajan, Fourier optimization and prime gaps, Com- ment. Math. Helv.94(2019), no. 3, 533–568

work page 2019