arxiv: 2604.25074 · v1 · submitted 2026-04-28 · 🧮 math.CA

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The smoothest average and some extremal problems for polynomials

Carlos Garz\'on, Jos\'e Gait\'an, Jos\'e Madrid

Pith reviewed 2026-05-07 14:25 UTC · model grok-4.3

classification 🧮 math.CA

keywords discrete derivativesaveraging kernelsextremal polynomialssharp constantsfinite differencesconvolutionFourier transform

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The pith

Symmetric kernels minimize the k-th discrete derivative norm of their convolution with any l2 function for k=3 and restricted higher orders.

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

The paper seeks to identify the symmetric kernel u with finite support and sum 1 that makes the smoothed function u*f as smooth as possible in the sense of minimizing the l2 norm of its k-th discrete gradient relative to the norm of f. For k=1 and 2 this was known, and the authors give new proofs using complex analysis. They solve the case k=3 without extra assumptions and the cases k=4 and 6 when the kernel's Fourier transform is non-negative. They also derive a general link between the optimal constants for k and for 2k. Sympathetic readers care because this pins down the best local averaging procedure for reducing higher-order variations in discrete data on the integers.

Core claim

The minimal value of sup ||∇^k (u * f)||_2 / ||f||_2 is attained by particular symmetric kernels for k=3, and for k=4,6 under the non-negative Fourier transform condition, providing the first such explicit results for k greater than 2, together with a relation connecting the sharp constants for ∇^k and ∇^{2k}.

What carries the argument

Extremal problems for polynomials that arise when optimizing the ratio involving higher-order differences after convolution, solved via complex analysis tools.

Load-bearing premise

Kernels must be symmetric, sum to exactly one, and for k=4 and 6 have non-negative Fourier transforms, as the optimality claims rely on these restrictions.

What would settle it

For k=3, exhibiting a symmetric kernel summing to 1 that yields a strictly smaller sup ratio than the identified one would disprove the claimed optimality.

Figures

Figures reproduced from arXiv: 2604.25074 by Carlos Garz\'on, Jos\'e Gait\'an, Jos\'e Madrid.

**Figure 1.** Figure 1: The optimal kernels u10 and u35 from Theorem 1.4. In order to prove Theorem 1.4, we apply the Fourier transform to reduce the problem to minimizing the norm ∥q∥L∞([−1,1]) over all polynomials q of degree at most n which have a double root at x = 1, satisfy q ′′(1) = 2, and do not take negative values on [−1, 1]. The extremal polynomial of this problem is obtained by manipulating properly the n-th Chebyshev… view at source ↗

**Figure 2.** Figure 2: Transformed Chebyshev polynomial T¯ 11(x). For each 1 ⩽ j ⩽ 4, let Θj (u) := Θj (u, k) be the j-th Jacobi Theta function with Jacobi elliptic modulus k, 0 < k2 < 1. Let Z(u) := Z(u, k) = Θ′ 4 (u)/Θ4(u) be the Jacobi Zeta function. Let cn(u) := cn(u, k), sn(u) := sn(u, k), and dn(u) := dn(u, k) be the Jacobi elliptic cosine, sine, and delta functions. Also, let K(k) and E(k) be the complete elliptic integra… view at source ↗

**Figure 3.** Figure 3: The optimal kernels u10 and u35 from Theorem 1.7 view at source ↗

**Figure 4.** Figure 4: Original data f and the smoothed data u35 ∗ f. when n → +∞. Here, k ∗ is the solution of K(k ∗ ) = 2E(k ∗ ). We have obtained numerically that k ∗ ≈ 0.9089 and K(k ∗ ) ≈ 2.3210 view at source ↗

**Figure 5.** Figure 5: Zolotarev polynomial Z211(x, k10, 1), with parameter choices k11 ≈ 0.911718 as in (1.10) and µ = 1 that will be discussed later. On the other hand, Kravitz and Steinerberger [6] showed that the sharp inequality for one derivative is closely related to the optimal inequality for two derivatives restricted to kernels with non-negative Fourier transform. Their argument suggests that this relation should hold … view at source ↗

**Figure 6.** Figure 6: Asymptotics of the value from Theorem 1.7, for 30 ⩽ n ⩽ 80. □ 5. Relation between the optimal constants and kernels associated to ∇k and ∇2k This section is devoted to establish Theorem 1.10. Our argument essentially relies on Fej´er-Riesz theorem to represent the Fourier transform of a kernel in Fn,2 as the square view at source ↗

read the original abstract

We study the problem of finding the "smoothest'' local average of a function $f \in \ell^2(\mathbb{Z})$ when we consider its convolution with suitable kernels $u$. The measurement of smoothness is as follows: Given a positive integer $k$, we aim to minimize the constant \begin{equation*} \sup_{0 \neq f \in \ell^2(\mathbb{Z})} \frac{\|\nabla^{k}(u\ast f)\|_{\ell^2(\mathbb{Z})}}{\|f\|_{\ell^2(\mathbb{Z})}} \end{equation*} among all symmetric kernels $u : \{-n,\dots,n\} \to \mathbb{R}$ with normalization $\sum_{j=-n}^{n}u(j) = 1$. We are also interested in finding the kernel for which the least constant is attained. For $k=1$ and $k=2$, the sharp constants and optimal kernels were obtained by Kravitz-Steinerberger, and Richardson. In this paper, we provide alternative proofs for $k\in \{1,2\}$ by using complex analysis tools. Moreover, we establish the case $k=3$, and also the cases $k\in \{4,6\}$ when the kernels are restricted to have non-negative Fourier transform. These are the first results in the literature for $k>2$. Finally, we deduce a general relation between the sharp constants and optimal kernels corresponding to $\nabla^k$ and $\nabla^{2k}$.

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

They deliver the first explicit results for k=3 outright and for k=4,6 under non-negative Fourier transform, plus a relation linking the constants for order k and 2k.

read the letter

The main thing to know is that this paper settles the k=3 case for the minimal constant in that sup norm of the third difference after convolution, and it gets the same for k=4 and 6 but only inside the subclass of kernels whose Fourier transform stays non-negative. It also gives fresh proofs for the k=1 and 2 cases via complex analysis and extracts a general relation between the sharp constants and kernels for ∇^k versus ∇^{2k}. Those are the first concrete results past k=2 in the literature they cite.

Referee Report

0 major / 3 minor

Summary. The paper studies the minimization, over symmetric finite-support kernels u summing to 1, of the operator norm sup ||∇^k (u ∗ f)||_2 / ||f||_2 for f in ℓ²(ℤ). It supplies complex-analytic proofs for the known cases k=1 and k=2, establishes the sharp constant and optimal kernel for k=3 without further restrictions, obtains the corresponding results for k=4 and k=6 under the additional constraint that the Fourier transform of u is non-negative, and derives a general relation linking the sharp constants and optimizers for ∇^k and ∇^{2k}.

Significance. If the derivations are correct, the work supplies the first explicit sharp constants and optimizers for k>2 in this discrete extremal problem, together with an alternative analytic approach to the low-order cases and a structural relation between the k and 2k problems. These results are of interest in discrete harmonic analysis and approximation theory; the explicit restrictions on the kernels are stated clearly and the claims are scoped accordingly.

minor comments (3)

The abstract and introduction should state explicitly whether the support size n of the kernel is fixed in advance or whether the minimization is taken over all n; the current wording leaves this ambiguous.
In the statement of the general relation between ∇^k and ∇^{2k} (presumably in the final section), the precise functional dependence between the two sharp constants should be written as an equation rather than described only in prose.
A short remark comparing the obtained constants for k=3 with the numerical values previously obtained by other methods (if any exist) would help the reader assess the result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The referee's summary accurately captures the scope and contributions of the work.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via complex analysis

full rationale

The paper supplies alternative proofs for the k=1 and k=2 cases using complex-analysis arguments that do not rely on prior fitted constants or self-referential definitions. It then establishes the k=3 case directly and the k=4,6 cases under the explicitly stated restriction of non-negative Fourier transforms. The final general relation between the sharp constants/optimal kernels for ∇^k and ∇^{2k} is deduced from these independently obtained results rather than presupposed. No step reduces by construction to a fitted parameter, a self-citation chain, or an ansatz smuggled in via prior work; all load-bearing claims are scoped to the stated kernel constraints and rest on direct analytic estimates.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard properties of the discrete Fourier transform on Z, symmetry of kernels, and tools from complex analysis (e.g., maximum principles or contour integrals) whose details are not supplied in the abstract.

axioms (2)

standard math Convolution with symmetric kernels preserves the l2 norm bounds in the stated sup expression.
Invoked implicitly when defining the smoothness constant.
domain assumption Fourier transform of the kernel controls the multiplier for the discrete derivative operator.
Central to relating the sup to the kernel's Fourier transform.

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discussion (0)

Reference graph

Works this paper leans on

4 extracted references

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[2]Bojanov, B., and Naidenov, N.On oscillating polynomials.J. Approx. Theory 162, 10 (2010), 1766–1787. [3]NIST Digital Library of Mathematical Functions.https://dlmf.nist.gov/, Release 1.2.6 of 2026- 03-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. ...

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[6]Kravitz, N., and Steinerberger, S.The smoothest average: Dirichlet, Fej´ er and Chebyshev. Bull. Lond. Math. Soc. 53, 6 (2021), 1801–1815. 24 J. GAIT ´AN, C. GARZ ´ON, AND J. MADRID [7]Lax, P. D.Proof of a conjecture of P. Erd¨ os on the derivative of a polynomial.Bull. Amer. Math. Soc. 50(1944), 509–513. [8]Lebedev, V. I.Zolotarev polynomials and extr...

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Fourier Anal

[13]Richardson, S.A Sharp Fourier Inequality and the Epanechnikov Kernel.J. Fourier Anal. Appl. 32, 1 (2026), Paper No

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[4]

[14]Steinerberger, S.Fourier uncertainty principles, scale space theory and the smoothest average. Math. Res. Lett. 28, 6 (2021), 1851–1874. (JG)Department of Mathematics, Virginia Polytechnic Institute and State Univer- sity, 225 Stanger Street, Blacksburg, V A 24061-1026, USA Email address:jogaitan@vt.edu (CG)Department of Mathematics, Virginia Polytech...

2021