Geometric properties of the Lebesgue function

Leokadia Bia{\l}as-Cie\.z; Mateusz Suder; Stefano De Marchi

arxiv: 2605.23299 · v1 · pith:MY5ROZSXnew · submitted 2026-05-22 · 🧮 math.NA · cs.NA

Geometric properties of the Lebesgue function

Leokadia Bia{\l}as-Cie\.z , Stefano De Marchi , Mateusz Suder This is my paper

Pith reviewed 2026-05-25 03:55 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords Lebesgue functiongeometric propertiesnumerical observationspolynomial interpolationintervalsquareopen problems

0 comments

The pith

Numerical computations reveal peculiar geometric behavior of the Lebesgue function on the interval and square.

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

The paper examines the Lebesgue function for polynomial interpolation and uses numerical experiments to map its shape on the interval from -1 to 1 and the square from -1 to 1 in two dimensions. It records a series of unexpected patterns in this geometry and lists several open questions about what the patterns mean exactly. A reader would care because the maximum of the Lebesgue function controls the worst-case error in interpolation, so its detailed shape could affect how nodes are chosen and how stable the approximations remain in practice.

Core claim

The Lebesgue function exhibits peculiar geometric behavior in the settings of the interval [-1,1] and the square [-1,1]^2, as revealed by numerical results. The authors supply these numerical observations and formulate several open problems related to the geometry of the Lebesgue function.

What carries the argument

The Lebesgue function, the pointwise maximum of the absolute sum of Lagrange basis polynomials, whose values determine the interpolation error bound.

If this is right

Observed patterns may constrain or suggest optimal choices of interpolation nodes on these domains.
The same geometric features could appear when the Lebesgue function is evaluated on other compact sets.
Resolving the listed open problems would give precise descriptions of extrema and level sets.
The numerical evidence provides a concrete starting point for theoretical proofs about the function's shape.

Where Pith is reading between the lines

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

If the patterns persist, they might be explained by symmetry properties of the underlying polynomial basis rather than by the specific nodes chosen.
Similar numerical scans on triangular or cubic domains could test whether the peculiar behavior is dimension-dependent.
Linking the observed geometry to the distribution of Lebesgue constants could tighten error estimates used in practice.

Load-bearing premise

The numerical computations performed accurately reflect the true continuous geometry of the Lebesgue function without significant discretization or rounding artifacts.

What would settle it

A computation at substantially higher resolution or an analytic construction that shows the reported geometric patterns fail to appear or change form in the continuous limit.

read the original abstract

We present a collection of observations concerning the peculiar behavior of the Lebesgue function in the setting of the interval $[-1,1]\subset \mathbb{R}$ and the square $[-1,1]^2\subset \mathbb{R}^2$. We provide numerical results and formulate several open problems related to the geometry of the Lebesgue function.

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

This is an exploratory numerical study that plots the Lebesgue function on the interval and square, notes some geometric patterns, and lists open questions without proving anything.

read the letter

The paper shows numerical visualizations of the Lebesgue function on [-1,1] and the square [-1,1]^2, highlighting some geometric features, and then poses several open problems about those features. It does a good job of documenting the observations without overstating them. The 2D plots in particular seem to reveal patterns that could be worth investigating further in bivariate interpolation. The authors are clear that this is meant to be a collection of observations rather than a complete theory. The soft spots are that the contribution is limited to these observations. There are no proofs, no new quantitative results, and the numerical details are not elaborated enough to allow independent verification of the reported behaviors. This makes the work more of a prompt for future research than a standalone advance. Since the paper advances no definitive claim, the lack of detailed numerics is not a major concern, but it does mean the value is mostly in the questions raised. This paper is for specialists in approximation theory who are interested in the Lebesgue function and might want to tackle the open problems. A reader looking for new methods or theorems won't find them here. I recommend against sending it for peer review. It lacks the substance to justify referee time in most journals, though it could be useful as an arXiv note to share the pictures and questions with the community.

Referee Report

1 major / 0 minor

Summary. The manuscript presents numerical observations concerning the peculiar geometric behavior of the Lebesgue function on the interval [-1,1] and the square [-1,1]^2, supplies numerical results to illustrate these behaviors, and formulates several open problems related to the geometry of the Lebesgue function.

Significance. If the reported observations hold under rigorous numerical verification, the work could serve as a useful exploratory study that identifies candidate geometric phenomena in approximation theory and motivates targeted theoretical investigations. As the manuscript advances no theorems, bounds, or definitive claims, its primary value would be in highlighting directions for future research rather than resolving specific questions.

major comments (1)

[Abstract] Abstract: the claim that numerical results reveal the peculiar geometric behavior is unsupported because the manuscript supplies no information on grid resolution, floating-point precision, or verification against known analytic cases for the Lebesgue function. This detail is load-bearing for the central observational content.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the feedback on the abstract. We respond to the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that numerical results reveal the peculiar geometric behavior is unsupported because the manuscript supplies no information on grid resolution, floating-point precision, or verification against known analytic cases for the Lebesgue function. This detail is load-bearing for the central observational content.

Authors: We agree that the abstract should be self-contained with respect to the numerical methodology. In the revised version we will add a sentence specifying the discretization (uniform grids with 2000 points on [-1,1] and 200 x 200 on the square), the use of double-precision arithmetic, and explicit verification on the Chebyshev extrema where the Lebesgue function is known analytically. The body of the manuscript already records these parameters; the revision will simply lift the essential information into the abstract. revision: yes

Circularity Check

0 steps flagged

No circularity: purely observational numerical study with open problems

full rationale

The paper reports numerical observations on Lebesgue function geometry in [-1,1] and [-1,1]^2 and formulates open problems. It advances no theorems, derivations, quantitative predictions, or fitted models. No load-bearing steps exist that could reduce to self-definition, fitted inputs renamed as predictions, or self-citation chains. The numerical results motivate questions rather than establish claims whose validity rests on the paper's own inputs. This matches the default expectation of no circularity for exploratory work without claimed derivations.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities; the work is purely observational.

pith-pipeline@v0.9.0 · 5577 in / 945 out tokens · 32016 ms · 2026-05-25T03:55:52.052906+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · 1 internal anchor

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L. Bos, M. Caliari, S. De Marchi, M. Vianello, Y. Xu,Bivariate Lagrange inter- polation at the Padua points: the generating curve approach, J. Approx. Theory 143 (2006), 15–25. 11 (a) Curvesℓ M P A (x, y) = 0for allA∈ M P5. The blue points indicate local maxima of the functionλM Pn (x, y) (b) Lebesgue function of the MP points of degree5. (c)Closeupofther...

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Caliari, S

M. Caliari, S. De Marchi, M. Vianello,Bivariate polynomial interpolation on the square at new nodal sets, Appl. Math. Comput. 165 (2005), 261-274

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T. A. Kilgore,A characterization of the Lagrange interpolation projections with minimal Tchebycheff norm, J. Approx. Theory 24 (1978), 273–288

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J. Liu, L. Zhu,Upper Bound for Lebesgue Constant of Bivariate Lagrange Inter- polation Polynomial on the Second Kind Chebyshev Points, J. Math. (2022), 5649969

work page 2022
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F. W. Luttmann, T. J. Rivlin,Some numerical experiments in the theory of polynomial interpolation, IBM J. Res. Develop. 9 (1965), 187–191

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E. Neuman,Projections in uniform polynomial approximations, Zastos. Mat. 14 (1974), 99–125

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T. J. Rivlin, The Chebyshev Polynomials, Wiley-Interscience, New York, 1974

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T. Tao,Local Bernstein theory, and lower bounds for Lebesgue constants, arXiv: 2603.21453, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
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http://mpmath.org/ 13

The mpmath development team.mpmath: a Python library for arbitrary-precision floating-point arithmetic (version 1.4.0), 2026. http://mpmath.org/ 13

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Beberok, L

T. Beberok, L. Bialas-Ciez, S. De Marchi,On the Lebesgue Constant of the Morrow–Patterson Points, Constr Approx 63 (2026), 31—65. 10 (a) Curvesℓ P ad A (x, y) = 0for allA∈ P ad4. The blue points indicate local maxima of the functionλP adn (x, y) (b) Lebesgue function of the Padua points of degree4. (c)Closeupoftheregionoutlined by the black square in (a)....

work page 2026

[2] [2]

Bloom,Erdös problems, https:/ /www.erdosproblems.com/

T. Bloom,Erdös problems, https:/ /www.erdosproblems.com/

work page

[3] [3]

de Boor, A

C. de Boor, A. Pinkus,Proof of the conjecture of Bernstein and Erdös concerning the optimal nodes for polynomial interpolation, J. Approx. Theory 24 (1978), 289–303

work page 1978

[4] [4]

L. Bos, M. Caliari, S. De Marchi, M. Vianello, Y. Xu,Bivariate Lagrange inter- polation at the Padua points: the generating curve approach, J. Approx. Theory 143 (2006), 15–25. 11 (a) Curvesℓ M P A (x, y) = 0for allA∈ M P5. The blue points indicate local maxima of the functionλM Pn (x, y) (b) Lebesgue function of the MP points of degree5. (c)Closeupofther...

work page 2006

[5] [5]

Brutman,On the Lebesgue function for polynomial interpolation, SIAM J

L. Brutman,On the Lebesgue function for polynomial interpolation, SIAM J. Numer. Anal. 15 (1978), 694–704

work page 1978

[6] [6]

Caliari, S

M. Caliari, S. De Marchi, M. Vianello,Bivariate polynomial interpolation on the square at new nodal sets, Appl. Math. Comput. 165 (2005), 261-274

work page 2005

[7] [7]

T. A. Driscoll, N. Hale, and L. N. Trefethen, editors, Chebfun Guide, Pafnuty Publications, Oxford, 2014. 12

work page 2014

[8] [8]

B. A. Ibrahimoglu,Lebesgue functions and Lebesgue constants in polynomial interpolation, J. Inequal. Appl. (2016), 2016:93

work page 2016

[9] [9]

T. A. Kilgore,A characterization of the Lagrange interpolation projections with minimal Tchebycheff norm, J. Approx. Theory 24 (1978), 273–288

work page 1978

[10] [10]

J. Liu, L. Zhu,Upper Bound for Lebesgue Constant of Bivariate Lagrange Inter- polation Polynomial on the Second Kind Chebyshev Points, J. Math. (2022), 5649969

work page 2022

[11] [11]

F. W. Luttmann, T. J. Rivlin,Some numerical experiments in the theory of polynomial interpolation, IBM J. Res. Develop. 9 (1965), 187–191

work page 1965

[12] [12]

Neuman,Projections in uniform polynomial approximations, Zastos

E. Neuman,Projections in uniform polynomial approximations, Zastos. Mat. 14 (1974), 99–125

work page 1974

[13] [13]

T. J. Rivlin, The Chebyshev Polynomials, Wiley-Interscience, New York, 1974

work page 1974

[14] [14]

Local Bernstein theory, and lower bounds for Lebesgue constants

T. Tao,Local Bernstein theory, and lower bounds for Lebesgue constants, arXiv: 2603.21453, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026

[15] [15]

http://mpmath.org/ 13

The mpmath development team.mpmath: a Python library for arbitrary-precision floating-point arithmetic (version 1.4.0), 2026. http://mpmath.org/ 13

work page 2026