Marcinkiewicz--Zygmund inequalities for scattered data on polygons

Hao-Ning Wu

REVIEW 3 major objections 5 minor 25 references

Marcinkiewicz--Zygmund inequalities for scattered data on polygons

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Pith's one-line read A quadrature rule built from scattered points on a polygon satisfies Marcinkiewicz–Zygmund inequalities for all 1≤p≤∞.

desk verdict A clean quadrature construction with a solid triangle part, but the polygon MZ theorem rests on a Markov inequality whose proof absorbs a degree-dependent constant, and the p=∞ branch has an invalid step. read the letter →

arxiv 2411.16584 v3 pith:TQQNZX7O submitted 2024-11-25 math.NA cs.NA

Hao-Ning Wu This is my paper · ORCID

classification math.NAcs.NA MSC 41A1741A0565D3242C15

keywords Marcinkiewicz–ZygmundinequalitiesBernstein–BézierpolynomialsquadratureinterpolationscatteredpointstrianglespolygonsMarkovinequality

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

Marcinkiewicz–Zygmund inequalities compare the $L^p$ norm of a polynomial with a weighted sum of its values at finitely many points, without requiring the quadrature rule to be exact for that polynomial. This paper claims that on any 2D polygon, any scattered point set that includes the vertices can be used as quadrature points: triangulate the polygon with those points as vertices and give each point one third of the sum of the areas of the triangles meeting there. For polynomials whose degree $N$ is small compared with the triangulation size, the resulting positive-weight quadrature satisfies these inequalities for every $1\le p\le\infty$, with constants depending only on $p$ and on the polygon's boundary. The payoff is that methods which only need norm control from discrete samples, such as discrete least-squares projection, can work on general polygonal domains without enforcing quadrature exactness. The paper also proves analogous estimates for 3-, 10-, and 21-point rules on a single triangle and supplies quadrature error bounds.

What carries the argument

Bernstein–Bézier polynomials, the basis of degree $d$ built from barycentric coordinates on a triangle, supply the norm equivalence, closed-form integrals, and interpolation error estimates that tie values at domain points to $L^p$ norms. The polygon argument is carried by the $d=1$ triangle rule, where the quadrature matrix is the identity and each vertex of a triangle receives weight $|T|/3$; summing these over the triangulation gives the weights in (3.11). The second load-bearing mechanism is Lemma 3.1, a Markov inequality on the polygon obtained by triangulating $\Omega$ with only its vertices: because those coarse triangles are fixed by the boundary, their shape parameters and the constant $C_5$ do not depend on $N$ or on the fine triangulation $\triangle$. This Markov inequality converts the second-derivative semi-norm $|\chi|_{p,2,\Omega}$ into $C_5 N^2 \|\chi\|_{p,\Omega}$, which is what turns the triangle-level estimates into the polygon inequalities.

What would settle it

Evaluate the left and right sides of (3.13) on a long thin quadrilateral whose triangulation uses only boundary and a few interior points, for $\chi(x,y)=x^N$ with $N$ just below $c^{-1}|\triangle|^{-2/p}$ and a range of $p$; if the relative discrepancy exceeds the claimed $\eta$ with constants as stated, the theorem cannot hold. Independently, maximizing the ratio in Lemma 3.1 over $P_N$ on the same polygon would expose whether its constant stays bounded as $N$ grows.

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Extended reading notes

Core claim

The paper establishes that scattered data on a polygon can directly define a quadrature rule with norm control. The construction triangulates the polygon with the scattered points as vertices, assigns each point weight $w_j = \frac13 \sum_{T\ni (x_j,y_j)} |T|$, and proves in Theorem 3.6 that for every polynomial $\chi$ of degree $N$ satisfying $N^2|\triangle|^{2/p}<1/c_1$ ($1\le p<\infty$), the discrete $L^p$ sum $\sum_j w_j |\chi(x_j,y_j)|^p$ differs from $\int_\Omega |\chi|^p$ by at most $\eta$ times the integral, with $\eta$ given explicitly; for $p=\infty$ the maximum over the scattered points is within relative error $\eta$ of the continuous maximum whenever $N$ is at most a constant times $\min\{1/|\triangle|, |\triangle|^{-1/2}\}\sqrt{\eta}$. The constants involve a polygon Markov constant that depends only on the boundary of $\Omega$, not on the fine triangulation, which is why the polygon setting succeeds where a single triangle with a fixed quadrature point set cannot. For triangles, the paper proves positive-weight Marcinkiewicz–Zygmund estimates for the 3-, 10-, and 21-point domain-point quadrature rules ($d=1,3,5$), exactness for all polynomials of degree at most $d$, and error bounds in $W^{d+1,p}$.

Load-bearing premise

The load-bearing premise is that on a fixed polygon, every derivative of a degree-$N$ polynomial is bounded by a constant times $N^2$ times the polynomial's norm, with the constant depending only on the polygon's boundary, not on $N$ or on how finely the polygon is triangulated; the proof gets this from the coarsest vertex-only triangulation and hinges on absorbing a degree-dependent factor from the triangle Markov inequality.

Editorial extensions

If this is right

In any polygonal domain, a positive-weight quadrature that is exact for linear polynomials and satisfies norm-control inequalities for higher-degree polynomials now exists for arbitrary scattered point sets containing the vertices.
The polygon Marcinkiewicz–Zygmund inequalities make it possible to run least-squares projection and related discretizations on scattered polygonal data without assuming quadrature exactness, since the discrete norm is controlled by the continuous norm whenever $N$ is below the stated threshold.
For a single triangle, scattered point sets containing the vertices inherit the polygon inequalities by applying the construction to the triangle itself (Corollary 3.1).
The triangle rules at $d=1,3,5$ provide positive-weight quadrature exact through degree $d$, and the associated error analysis bounds the quadrature error for $f\in W^{2,p}$ on polygons by $K_1 C_7 \max_T |A_T|^{1/q} |f|_{2,p,\Omega}$.
If the Markov inequality on polygons is sharp enough, the same quadrature weights give uniform bounds that do not degrade as the scattered points are refined, provided the degree $N$ is kept within the stated size condition.

Reading between the lines

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

A reader could take the boundary-only Markov constant and build polygon hyperinterpolation schemes whose discrete inner products use these scattered-point weights; the paper does not carry out that application.
If the positivity pattern seen for $d=1,3,5$ continues for all odd $d$, the same coefficient-free construction would yield positive-weight triangle rules at every odd degree; this is a testable extension, not a claim in the paper.
One could replace the 3-point triangle rule inside each coarse triangle by the 10- or 21-point rule to raise the exactness degree of the polygon quadrature, at the cost of requiring interior domain points in every triangle.

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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: the construction is clean and the triangle part is solid, but the main theorem on polygons is not proved as written. The stress-test's central criticism is correct: Lemma 3.1's proof absorbs a degree-dependent constant into a boundary-only constant, and the p=∞ branch has an additional invalid estimate.

What's new: the paper takes the standard vertex-area quadrature rule (3.10)-(3.11) and claims it satisfies Marcinkiewicz–Zygmund inequalities on polygons for scattered points. The idea of using the coarsest triangulation to get a Markov inequality on the polygon is good; it sidesteps the shape-parameter blow-up that blocks the single-triangle case. The exactness for P1, the error bound in Theorem 3.5, and the triangular MZ estimates for d=1,3,5 are all sound. The numerical experiments match the analysis.

Where it falls apart: Lemma 3.1 states that there is a constant C5 depending only on the boundary with |χ|_{p,2,Ω} ≤ C5 N^2 ||χ||_{p,Ω}. The proof applies Lemma 2.3 on the coarse triangulation. That lemma's constant is K0(2d)^{α+β}, with K0 depending on d. For the Bernstein basis, K0 is not uniform in d: coefficients c_k = (-1)^k C(d,k) give the polynomial (1-2t)^d, sup norm 1, but max |c_k| ~ 2^d/√d. So C5 cannot be boundary-only in general. Without a uniform K0, c1 in Theorem 3.6 depends on N, and the hypothesis N^2|Δ|^{2/p}<1/c1 is not the stated boundary-only condition. This affects both p<∞ and p=∞ branches.

The p=∞ branch has a separate gap: it bounds the first-order seminorm |χ|_{∞,1} by |Δ||χ|_{∞,2}, which is false (a linear polynomial is a counterexample). A first-order Markov inequality on the polygon would repair this, but that's not what is written.

Even after repair, the theorem is weaker than the classical MZ form: the constants grow with the number of triangles, and on quasi-uniform meshes the degree is restricted to N ~ m^{1/(2p)} for p<∞ and N ~ m^{1/4} for p=∞. For p>1 that is well below the usual N ~ m^{1/2}, so the MZ constants are not uniform in the number of points. The paper should state this limitation explicitly.

Bottom line: the construction is worth knowing and the triangle part is a solid contribution, but the polygon theorem needs a real rewrite. The paper deserves refereeing, with instructions to focus on Lemma 3.1 and the p=∞ case.

Referee Report

3 major / 5 minor

Summary. The paper proposes quadrature rules for triangles and polygons built from Bernstein–Bézier polynomials and scattered points. On a triangle, the rule uses domain points with weights defined through column sums of the inverse Bernstein–Bézier interpolation matrix; the paper proves exactness for polynomials of degree d and derives Marcinkiewicz–Zygmund estimates for d = 1, 3, 5, plus error estimates for d ≥ 1. On a polygon, the scattered points are used as vertices of a triangulation, the quadrature weights are one third of the sum of adjacent triangle areas, and the paper claims exactness for linear polynomials, an O(|△|^2) error bound, and Marcinkiewicz–Zygmund inequalities for 1 ≤ p ≤ ∞. Numerical experiments on a triangle and a Georgia-shaped polygon illustrate quadrature accuracy.

Significance. If the main theorem were established, the paper would provide an explicit, positive-weight quadrature rule for scattered data on polygons with Marcinkiewicz–Zygmund inequalities, a practically useful extension of the sphere and manifold results. The triangle part is genuinely explicit: the weights are given in closed form, the quadrature rule is exact for P_d, and the constants are not fitted. The numerical experiments support the quadrature error claims. However, the central polygon result is not established: the proof relies on a Markov inequality with a degree-uniform constant that is false as stated, and on geometric bounds whose constants are not controlled by the boundary alone. The paper therefore needs substantial revision before its main claim can be accepted.

major comments (3)

[§3.2, Lemma 3.1 (eq. (3.12))] Lemma 3.1 is false as stated, and the proof does not provide the degree-uniform constant it claims. Let [a,b] be the projection of Ω on the x-axis and set P_d(x,y) = T_d(2(x-a)/(b-a)-1). Then P_d ∈ P_d and ||P_d||_{∞,Ω} ≤ 1, while ||∂_x P_d||_{∞,Ω} = 2 d^2/(b-a). Lemma 3.1 with α+β=1 would require a boundary-only C5 satisfying C5 d ≥ c d^2 for all d, which is impossible. The proof of Lemma 3.1 absorbs the degree-dependent constant K0(d) from Lemma 2.1 into C5, but K0(d) is not shown to be uniformly bounded and in the Bernstein basis it grows with d. This is load-bearing: equations (3.18)–(3.19) and the p=∞ branch of Theorem 3.6 use Lemma 3.1 to replace |χ|_{p,2,Ω} by C5 N^2 ||χ||_{p,Ω}, so the stated condition N^2|△|^{2/p} < 1/c1 with c1 boundary-only is unsupported. A correct Markov estimate changes the admissible N-|△| relation in Theorem 3.6.
[§3.2, Lemma 3.2 and Theorem 3.6 constants] The constant C7 in Lemma 3.2 is not determined by the boundary of Ω. The proof defines γ△ = |△|/ρ_{Tmin}; this ratio depends on the triangulation and can be arbitrarily large for a fixed polygon, for example in triangulations containing very thin triangles with longest edge of order one and arbitrarily small inradius. The sentence in the proof asserting that γ△ depends on the boundary of Ω is therefore incorrect, and the inequality |#△|^{1-1/p}|△|^2 ≤ C7|△|^{2/p} is not valid uniformly over all triangulations of a fixed polygon. This affects the error bound in Theorem 3.5 and the constants c1,c2 in Theorem 3.6. Relatedly, Lemma 2.2 states that for 1 ≤ p < ∞ the constant K1 depends on κ_T, but in the proof of Theorem 3.6 K1 is declared to depend only on p 'since d=1'; without a shape-regularity assumption on the triangulation this is not justified.
[§3.2, proof of Theorem 3.6(b), p=∞ branch] The step bounding the vertex value near an extremum is not justified. The proof writes |χ(x*,y*) - χ(x_J,y_J)| ≤ 2|T2||χ|_{∞,1,T2} and then passes from ‖χ‖_{∞,Ω} - K1|T2|^2|χ|_{∞,2,T2} - 2|T2||χ|_{∞,1,T2} to ‖χ‖_{∞,Ω} - C4|△||χ|_{∞,2,T2}. This implicitly requires a bound of the first-order seminorm |χ|_{∞,1,T2} by the second-order seminorm |χ|_{∞,2,T2}, which is false for polynomials: for example, a nonconstant linear polynomial has zero second seminorm but nonzero first seminorm. The argument should instead use a first-order Markov estimate on the polygon; as written, the p=∞ branch contains a genuine gap.

minor comments (5)

[§3.2, Theorem 3.5 proof] The proof states '|#△||△|^2 ≤ C7', but the estimate available from Lemma 3.2 is |#△|^{1-1/p}|△|^2 ≤ C7|△|^{2/p} (equivalently |#△|^{1/q}|△|^2 ≤ C7|△|^{2/p}); the displayed line should be corrected.
[§2, Lemma 2.3] The citation '[16, Theorems 2.32]' should be '[16, Theorem 2.32]'.
[Throughout] There are several typos: 'Combing both inequalities' should be 'Combining both inequalities'; 'inscribed circle in T' should be 'inscribed in T'; 'out arguments' in the final remark should be 'our arguments'; and 'BT(ξ010))' has an extra parenthesis.
[§4] The numerical experiments test quadrature exactness and integration error only; they do not directly test the Marcinkiewicz–Zygmund inequalities asserted in Theorem 3.6, so the claim that they 'validate our construction' is stronger than what is demonstrated.
[§3.3, Theorem 3.3 proof] The p=∞ proof contains a mismatched delimiter, '|χ‖_{∞,T}', which should be '‖χ‖_{∞,T}'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the quadrature construction and MZ proof are self-contained; the flagged Markov-constant uniform-bound issue is a proof gap, not a circular reduction.

full rationale

The derivation chain is self-contained. Theorem 3.1 defines the triangle quadrature weights by exactness, w = A_T / binom(d+2,2) times the column sums of B^{-1}, and Theorem 3.3 derives the triangle Marcinkiewicz–Zygmund estimates directly from the Bernstein norm-equivalence Lemma 2.1 and the interpolation matrix, without assuming the target inequality. The polygon rule (3.10)-(3.11) is simply the d=1 triangle rule summed over a triangulation, and Theorem 3.5 follows from Theorems 3.1-3.2 and Lemma 3.2. Theorem 3.6 assembles the per-triangle estimates, uses the mesh relation (2.1), Lemma 3.2, and Lemma 3.1's Markov bound; none of these steps assumes (3.13) or (3.14). The external inputs [15,16] are standard interpolation and Markov inequalities with stated constants, and the self-citations [1-4,24] are background applications in the introduction, not load-bearing premises. No parameter is fitted to the target quantity, and no prediction is a renamed input. The reviewer-flagged concern about Lemma 3.1 — that its proof absorbs K0(d) from Lemma 2.3 into a boundary-only constant C5 without proving K0 is bounded uniformly in d, and that Lemma 2.2's K1 for p<∞ may depend on the triangle shape κ_T — is a possible correctness gap in the stated constants, not a circularity. Even if valid, it would make the condition N^2|△|^{2/p} < 1/c1 unsupported, but it would not make Theorem 3.6 equivalent to its own assumptions.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard Bernstein-Bezier analysis and on one unproved degree-independent Markov inequality on polygons; no free parameters or invented entities are used.

assumptions (4)

standard math Standard Bernstein-Bezier basis properties, norm equivalence, and Markov inequality on triangles as stated in [16].
Used throughout Section 3; in particular Lemma 2.3 with K2 = K0(2d)^{alpha+beta} supplies the Markov bound that Lemma 3.1 builds on.
domain assumption The norm equivalence constant K0 in Lemma 2.1 remains bounded independent of the polynomial degree d.
Lemma 3.1 claims a Markov constant C5 depending only on the boundary of Omega; its proof absorbs K0, which Lemma 2.1 says depends on d, so the degree-independence is assumed rather than proved.
domain assumption The scattered points admit a non-degenerate triangulation of Omega with these points as vertices.
The construction and Theorem 3.6 start from such a triangulation; no degeneracy handling is discussed.
ad hoc to paper Lemma 3.1: a Markov inequality on the polygon with constant depending only on the boundary of Omega.
This is the load-bearing premise for bounding the Sobolev semi-norms in Theorem 3.6; its proof is only sketched and the degree-independence is not established.

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Cite this review

Pith. "Pith review of Marcinkiewicz--Zygmund inequalities for scattered data on polygons." pith.science (2026). https://pith.science/paper/TQQNZX7O

@misc{pith2026241116584,
  author       = {Pith},
  title        = {Pith review of: Marcinkiewicz--Zygmund inequalities for scattered data on polygons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TQQNZX7O}},
  note         = {Machine review of arXiv:2411.16584}
}

abstract

Given a set of scattered points on a regular or irregular 2D polygon, we aim to employ them as quadrature points to construct a quadrature rule that establishes Marcinkiewicz--Zygmund inequalities on this polygon. The quadrature construction is aided by Bernstein--B\'{e}zier polynomials. For this purpose, we first propose a quadrature rule on triangles with an arbitrary degree of exactness and establish Marcinkiewicz--Zygmund estimates for 3-, 10-, and 21-point quadrature rules on triangles. Based on the 3-point quadrature rule on triangles, we then propose the desired quadrature rule on the polygon that satisfies Marcinkiewicz--Zygmund inequalities for $1\leq p \leq \infty$. As a byproduct, we provide error analysis for both quadrature rules on triangles and polygons. Numerical results further validate our construction.

Figures

Figures reproduced from arXiv: 2411.16584 by the authors.

**Figure 2.** Locations of quadrature points, corresponding quadra [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

An and H.-N

C. An and H.-N. Wu , On the quadrature exactness in hyperinterpolation , BIT, 62 (2022), pp. 1899–1919, https://doi.org/10.1007/s10543-022-00935-x

doi:10.1007/s10543-022-00935-x 2022

[2] [2]

An and H.-N

C. An and H.-N. Wu , Bypassing the quadrature exactness assumption of hyperinterpolation on the sphere , J. Complexity, 80 (2024), p. No. 101789, https://doi.org/10.1016/j.jco.2023.101789

arXiv 2024

[3] [3]

An and H.-N

C. An and H.-N. Wu , Is hyperinterpolation eﬃcient in the approximation of sin- gular and oscillatory functions? , J. Approx. Theory, 299 (2024), p. No. 106013, https://doi.org/10.1016/j.jat.2023.106013

work page arXiv 2024

[4] [4]

Spherical configurations and quadrature methods for integral equations of the second kind

C. An and H.-N. Wu , Spherical conﬁgurations and quadrature methods for integr al equations of the second kind , arXiv preprint arXiv:2408.14392, (2024)

work page Pith review arXiv 2024

[5] [5]

B ¨ottcher, S

A. B ¨ottcher, S. Kunis, and D. Potts , Probabilistic spherical Marcinkiewicz- Zygmund inequalities , J. Approx. Theory, 157 (2009), pp. 113–126, https://doi.org/10.1016/j.jat.2008.07.006

work page doi:10.1016/j.jat.2008.07.006 2009

[6] [6]

C. K. Chui and L. Zhong , Polynomial interpolation and Marcinkiewicz- Zygmund inequalities on the unit circle , J. Math. Anal. Appl., 233, pp. 387–405, https://doi.org/10.1006/jmaa.1999.6337

work page arXiv 1999

[7] [7]

Dai and H

F. Dai and H. W ang , Positive cubature formulas and Marcinkiewicz- Zygmund inequalities on spherical caps , Constr. Approx., 31 (2010), pp. 1–36, https://doi.org/10.1007/s00365-009-9041-7

work page doi:10.1007/s00365-009-9041-7 2010

[8] [8]

De Marchi and A

S. De Marchi and A. Kro ´o, Marcinkiewicz-Zygmund type results in multivariate domai ns, Acta Math. Hungar., 154 (2018), pp. 69–89, https://doi.org/10.1007/s10474-017-0769-4

work page doi:10.1007/s10474-017-0769-4 2018