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arxiv: 2606.23409 · v1 · pith:WVWT73CJnew · submitted 2026-06-22 · 🧮 math.NA · cs.NA

When do perturbed Chebyshev--Lobatto points remain Chebyshev?

Hao-Ning Wu This is my paper

Pith reviewed 2026-06-26 07:28 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Chebyshev-Lobatto pointsLebesgue constantperturbation stabilitypolynomial interpolationangular perturbationnumerical stability
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The pith

Perturbed Chebyshev-Lobatto points retain logarithmic Lebesgue constants when n σ_n (log n + 1) is bounded by a small constant.

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

This paper determines how much Chebyshev-Lobatto interpolation nodes can be moved while preserving their characteristic logarithmic Lebesgue constants. It proves a deterministic bound: when the product of the degree n, the maximum angular perturbation σ_n, and (log n + 1) stays below a small threshold, the Lebesgue constant remains O(log n). A reader would care because these points underpin stable high-degree polynomial approximation in many numerical schemes, and the result quantifies their robustness to placement errors that arise in practice. The work also shows that perturbations of size 1/n are too large to preserve the property uniformly.

Core claim

For nodes x_j = cos(j π / n + ε_j) with |ε_j| ≤ σ_n, if n σ_n (log n + 1) is bounded by a sufficiently small constant, then the Lebesgue constant of the associated interpolation operator remains logarithmic in n. The argument proceeds by mapping to the angular variable, expressing the Lebesgue function via trigonometric polynomials, and applying Bernstein's inequality to control its growth under the stated perturbation size.

What carries the argument

Cosine parametrization of the nodes together with Bernstein's inequality for trigonometric polynomials, which directly bounds the Lebesgue function when perturbations are controlled in the angular variable.

If this is right

  • Analytic interpolation inside Bernstein ellipses remains free of Runge-type divergence when the perturbation condition holds.
  • Pseudospectral differentiation errors remain controlled in the regime where the Lebesgue constant stays logarithmic.
  • Uniform perturbations of angular size 1/n produce a worst-case loss of the logarithmic Lebesgue constant.
  • The transition from stable to unstable Lebesgue constants occurs near n σ_n of order 1 over log n, consistent with the reported numerical experiments.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Limited adaptive adjustment of nodes could be performed while keeping the interpolation operator stable, provided the angular shifts respect the derived bound.
  • Because the analysis is angular, physical-space perturbations near the endpoints may be larger than those near the center without violating the condition.
  • The same angular technique might be tested on other clustered node families, though the required inequalities would differ.

Load-bearing premise

Perturbations are measured uniformly in the angular cosine variable so that Bernstein's inequality for trigonometric polynomials applies directly.

What would settle it

A family of perturbations with n σ_n (log n + 1) bounded but with the maximum of the Lebesgue function growing faster than any fixed multiple of log n would disprove the stability claim.

Figures

Figures reproduced from arXiv: 2606.23409 by Hao-Ning Wu.

Figure 1
Figure 1. Figure 1: Numerical motivation. The colour indicates log [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Lebesgue constants for perturbed Chebyshev–Lobat [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Lebesgue functions for selected node sets at fixed [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Interpolants of the Runge function f(x) = 1/(1 + 25x 2 ). For the smallest degree shown, the equally spaced interpolant is included to display the onset of the classical endpoint oscillation. For the larger degrees, the comparison focuses on the Chebyshev–Lobatto interpolant and an angular perturbation with σn = (n(log n + 1))−1 . 14 [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
read the original abstract

Chebyshev points are distinguished in polynomial interpolation by the logarithmic growth of their Lebesgue constants. This paper asks a simple question: how much can Chebyshev points be perturbed before they cease to behave like Chebyshev points? We study perturbed Chebyshev--Lobatto nodes $x_j=\cos(j\pi/n+\varepsilon_j)$, with angular perturbations $|\varepsilon_j|\leq \sigma_n$. The study is motivated by numerical experiments showing a broad stable region when the mesh fraction $n\sigma_n$ is small and rapid amplification for larger perturbations; the observed transition region is consistent with the curve $n\sigma_n\asymp(\log n)^{-1}$. The main result is a deterministic worst-case stability estimate: if $n\sigma_n(\log n+1)$ is bounded by a sufficiently small constant, then the Lebesgue constant remains logarithmic. The proof uses the cosine parametrization and Bernstein's inequality for trigonometric polynomials, thereby exploiting the angular geometry of the Chebyshev--Lobatto grid rather than a Markov inequality in the physical variable. We also give a worst-case obstruction at the angular mesh scale, showing that perturbations of order $1/n$ cannot be allowed uniformly. Consequences are derived for analytic interpolation in Bernstein ellipses, for the absence of Runge-type divergence in the stable analytic regime, and for pseudospectral differentiation. Numerical experiments illustrate the transition in the Lebesgue constants, the shape of the associated Lebesgue functions, Runge-function interpolants, and finite-precision differentiation errors.

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.

Referee Report

0 major / 1 minor

Summary. The manuscript studies perturbed Chebyshev-Lobatto nodes x_j = cos(j π / n + ε_j) with angular perturbations |ε_j| ≤ σ_n. It proves a deterministic stability result: if n σ_n (log n + 1) is bounded by a sufficiently small constant, the associated Lebesgue constant remains O(log n). The argument maps to the angular variable and applies Bernstein's inequality for trigonometric polynomials. A matching obstruction is shown at the scale σ_n ∼ 1/n. Consequences are derived for interpolation in Bernstein ellipses, absence of Runge divergence, and pseudospectral differentiation, with numerical experiments illustrating the transition.

Significance. If the central estimate holds, the work supplies a sharp, parameter-free characterization of the perturbation tolerance that preserves the logarithmic Lebesgue constant of Chebyshev-Lobatto points. The angular parametrization together with Bernstein's inequality yields a clean derivation that directly multiplies the classical log n factor by n σ_n; the obstruction result at mesh scale 1/n confirms that the scaling is tight. These features, together with the applications to analytic regimes and differentiation, make the contribution useful for robustness analysis in spectral methods.

minor comments (1)
  1. [Abstract] Abstract: the transition curve is stated as n σ_n ≍ (log n)^{-1} while the theorem uses the product n σ_n (log n + 1); a short clarifying sentence relating the two forms would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via trigonometric inequalities

full rationale

The central result is a deterministic bound: when n σ_n (log n +1) is sufficiently small, the Lebesgue constant for perturbed Chebyshev-Lobatto nodes x_j = cos(j π /n + ε_j) remains O(log n). The proof maps to the angular variable and applies Bernstein's inequality for trigonometric polynomials to control the growth of the Lebesgue function relative to the equispaced trigonometric case. This produces the factor n σ_n multiplied by the known logarithmic term for unperturbed points. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation. The obstruction at σ_n ~ 1/n is likewise obtained from the same scaling. The derivation is independent of the motivating numerical experiments and relies only on standard external inequalities.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on classical approximation-theory tools with no free parameters fitted to data and no new postulated entities.

axioms (1)
  • standard math Bernstein's inequality for trigonometric polynomials
    Invoked to control growth after the cosine reparametrization of the perturbed nodes.

pith-pipeline@v0.9.1-grok · 5792 in / 1292 out tokens · 40614 ms · 2026-06-26T07:28:15.580001+00:00 · methodology

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