Semi-discrete Gruss-Voronovskaya-type and Gruss-type estimates for Bernstein-Kantorovich polynomials

Sorin G. Gal

REVIEW 2 major objections 3 minor 11 references

Semi-discrete Gruss-Voronovskaya-type and Gruss-type estimates for Bernstein-Kantorovich polynomials

T0 review · 2 major / 3 minor · reviewed 2026-08-27 · deepseek-v4-flash

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Pith's one-line read For Bernstein–Kantorovich polynomials, the gap $K_n(fg)-K_n(f)K_n(g)$ obeys an explicit semi-discrete pointwise bound, with $O(1/n)$ decay and a Voronovskaya-type limit.

desk verdict Competent but very thin: the paper applies the author's own semi-discrete Voronovskaya theorem to Bernstein–Kantorovich polynomials, and the proof's soundness depends entirely on an unstated, self-cited corollary. read the letter →

arxiv 2010.02697 v1 pith:Y6XQA7NY submitted 2020-09-25 math.CA

Sorin G. Gal This is my paper · ORCID

classification math.CA MSC 41A3641A2541A60

keywords Bernstein–Kantorovichpolynomialssemi-discreteGrüss–Voronovskaya-typeestimateperturbedGrüss-typemodulusofcontinuitydivideddifferencespositivelinearoperatorsVoronovskayatheoremGrüssinequality

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The pith

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

The reading

This note proves a semi-discrete Grüss–Voronovskaya-type estimate for the Bernstein–Kantorovich polynomials $K_n$. The main theorem bounds, for all $f,g\in C^2[0,1]$, $n\in\mathbb{N}$ and $x,y\in[0,1]$ with $x\neq y$, the pointwise Grüss defect $K_n(fg)(x)-K_n(f)(x)K_n(g)(x)$ by an explicit expression built from divided differences, moduli of continuity of second derivatives, and the operator's own approximation errors. Because the estimate keeps a divided difference between $x$ and $y$ rather than passing to the derivative limit, it acts as a bridge between discrete Grüss-type inequalities and the asymptotic Voronovskaya regime. As consequences, the paper obtains a perturbed Grüss-type estimate, uniform $O(1/n)$ decay of the defect, and, after scaling by $n$ and letting $y\to x$, the classical limit $n[K_n(fg)(x)-K_n(f)(x)K_n(g)(x)]\to x(1-x)f'(x)g'(x)$.

What carries the argument

The load-bearing element is an algebraic identity expressing the corrected Grüss defect as a combination of three semi-discrete Voronovskaya remainders $R_n(h)=K_n(h)(x)-h(x)-\alpha[x,y;h]-E_n(x,y)h''(x)/2$, where $\alpha=(1-2x)/(2(n+1))$ and $E_n(x,y)=F_n(x)+(x-y)(1-2x)/(2(n+1))$, plus the product $[K_n(f)(x)-f(x)][g(x)-K_n(g)(x)]$. The paper bounds each remainder by invoking Corollary 4.1 of [4], a general semi-discrete quantitative Voronovskaya-type theorem for positive linear operators, with the explicitly chosen $E_n$ and $F_n$; the constants $1/3$ and $2\sqrt6$ enter through that corollary.

What would settle it

Choose $f(t)=g(t)=t^2$, fix $x=0.3$, $y=0.7$, $n=10$, compute every term in Theorem 2.1 directly from the Bernstein–Kantorovich formula, and check whether the left-hand side exceeds the right-hand side; a single violation would disprove the theorem. Equivalently, test the underlying semi-discrete Voronovskaya remainder $R_n(t^2)$ against the claimed modulus bound to see whether the asserted constants $1/3$ and $2\sqrt6$ hold.

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Extended reading notes

Core claim

The central claim is Theorem 2.1: for all $f,g\in C^2[0,1]$, $n\in\mathbb{N}$, and $x,y\in[0,1]$ with $x\neq y$, the quantity $$\left|K_n(fg)(x)-K_n(f)(x)K_n(g)(x)+\frac{(x-y)(1-2x)}{2(n+1)}([x,y;f][x,y;g]-f'(x)g'(x))-F_n(x)f'(x)g'(x)\right|$$ is bounded above by $$\left[\frac{1}{(n+1)^2}\left(x(1-x)(n-1)+\frac13\right)+\frac{|x-y|}{\sqrt{3}\,\sqrt{n+1}}\right]\Bigl[\omega_1\Bigl((fg)'';|x-y|+\frac{2\sqrt6}{\sqrt{n+1}}\Bigr)+\|g\|\,\omega_1\Bigl(f'';|x-y|+\frac{2\sqrt6}{\sqrt{n+1}}\Bigr)+\|f\|\,\omega_1\Bigl(g'';|x-y|+\frac{2\sqrt6}{\sqrt{n+1}}\Bigr)\Bigr]+|K_n(f)(x)-f(x)|\,|K_n(g)(x)-g(x)|.$$ Here $[x,y;f]=(f(x)-f(y))/(x-y)$ is the divided difference, $\omega_1$ is the usual modulus of continuity, and $F_n(x)=\frac{1}{(n+1)^2}(x(1-x)(n-1)+\frac13)$. The substantive discovery is that the Grüss defect can be decomposed exactly into three copies of the same semi-discrete Voronovskaya remainder, so a single general estimate from prior work yields a fully explicit three-term error bound.

Load-bearing premise

The estimate rests entirely on the author's earlier general theorem (Corollary 4.1 of [4]) being correct and actually applying to the Bernstein–Kantorovich operator with the stated correction terms; this note neither restates that theorem nor verifies the moment computations fixing the constants $1/3$ and $2\sqrt6$.

Editorial extensions

If this is right

For $f,g\in C^2[0,1]$, the Grüss defect satisfies $\|K_n(fg)-K_n(f)K_n(g)\|=O(1/n)$ uniformly in $x$, as stated in Remark 2.3.
For $f,g\in C^3[0,1]$, the scaled defect obeys $n[K_n(fg)-K_n(f)K_n(g)]\to e_1(1-e_1)f'g'$ in the uniform norm with error $O(1/\sqrt{n})$, recovering the classical Grüss-Voronovskaya rate from [1], as stated in Remark 2.2.
The main bound is genuinely two-point: it holds for every $y\neq x$, so $y$ can be chosen adaptively (for example depending on $n$) without a new proof.
Removing the correction terms in Theorem 2.1 yields a perturbed Grüss-type estimate whose extra term is $|F_n(x)|\,|f'(x)g'(x)|=O(1/n)$, which becomes negligible as $y\to x$ (Remark 2.3).
The same algebraic decomposition applies to any positive linear operator for which a semi-discrete Voronovskaya estimate is available, so the method of [4] yields analogous Grüss-Voronovskaya and Grüss-type estimates for other families (Remark 2.4).

Reading between the lines

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

The constants $1/3$ and $2\sqrt6$ are inherited from the cited general corollary and are not optimized here; a direct calculation of the moments of $K_n$ could confirm or improve them without disturbing the $O(1/n)$ rate.
The bound leaves $y$ as a free parameter; choosing $y=x\pm 1/\sqrt{n+1}$ would balance the factor $|x-y|$ against the modulus arguments, likely giving cleaner uniform estimates, but the paper does not explore this.
For $f,g$ with third or fourth derivatives, the same identity should generate higher-order corrections in the scaled defect (terms of order $1/n^2$), refining the $O(1/\sqrt{n})$ rate; this extension is not stated in the paper.

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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

Quick take: this is a legitimate but very incremental note. What's new is an explicit pointwise semi-discrete Grüss–Voronovskaya-type inequality for Bernstein–Kantorovich polynomials, with constants, plus a perturbed Grüss-type estimate. The algebraic decomposition in the proof is correct—I checked the moment bookkeeping—and the result does recover the known asymptotics, which is a useful sanity check. Credit is due for making the bound fully quantitative, and the Grüss-type consequence is a neat observation.

The soft spot is obvious and real: the proof is just an application of Corollary 4.1 from the author's own paper [4], and that corollary is never stated. All the quantitative content—the 1/3, the 1/√3, the 2√6, the exact form of E_n—enters through that imported result. The note doesn't verify that Bernstein–Kantorovich operators satisfy the hypotheses, and it doesn't show the moments that justify the constants. If Corollary 4.1 has different coefficients or extra terms, Theorem 2.1 as stated is not established. That is a genuine self-containedness problem, not a manufactured one. It is also not a circularity; the theorem is a new specialization, not a restatement, so the self-citation burden is low.

Another minor issue: no numerical examples, no discussion of sharpness, and the application in Remark 2.2 just recovers an existing order of approximation. Still, for a note this length, that's acceptable if the imported theorem is sound.

Who is this for? Specialists in positive linear operators and quantitative Voronovskaya theory. It could be useful as a citable example of the semi-discrete machinery. My recommendation: it deserves a serious referee, because the underlying idea is plausible and the algebraic core is correct, but the author should be required to state Corollary 4.1 in full and explicitly check its hypotheses. If that corollary checks out, it's publishable in a standard approximation-theory journal.

Referee Report

2 major / 3 minor

Summary. The paper claims a pointwise, quantitative Grüss-Voronovskaya-type estimate for Bernstein-Kantorovich polynomials. Theorem 2.1 bounds an expression involving K_n(fg)-K_n(f)K_n(g), divided differences of f and g, and a term proportional to f'(x)g'(x), by a combination of moduli of continuity of second derivatives and the residual terms |K_n(f)(x)-f(x)| and |K_n(g)(x)-g(x)|. The proof consists of an algebraic rewriting of the target expression followed by three applications of Corollary 4.1 from the author's previous paper [4]. Remarks 2.2 and 2.3 draw asymptotic Grüss-Voronovskaya and Grüss-type consequences from the main estimate.

Significance. If correctly established, Theorem 2.1 would be a useful explicit pointwise estimate for Bernstein-Kantorovich polynomials, with the attractive feature that for linear functions f and g the estimate is an equality. The note is concise and the claimed rate O(1/n) in Remark 2.3 is plausible. However, the proof has a concrete algebraic error and relies entirely on an unstated corollary of a self-cited paper, so the published result is not currently verifiable.

major comments (2)

[Proof of Theorem 2.1] The displayed equality in the proof, after 'using Corollary 4.1 in [4], we obtain', is false as an algebraic identity. For example, take n=1, x=1/4, y=0, and f(t)=g(t)=t^2. Then the left-hand side inside the absolute value equals 107/5760, while the right-hand side, namely the expression with the three bracket terms and the residual product, equals 833/23040, which is larger by 9/512. The subsequent bound is applied to that right-hand side, not to the theorem's left-hand side. Since the extra term is a(1-(x-y))(f'(x)g'(x)-[x,y;f][x,y;g]), it does not vanish generally. This invalidates the proof of Theorem 2.1 as written.
[Proof of Theorem 2.1, use of Corollary 4.1 of [4]] The decisive quantitative content is imported from Corollary 4.1 of [4], but that corollary is never stated and no verification of its hypotheses for the Bernstein-Kantorovich operator K_n is provided. In particular, the constants 1/3, 1/sqrt(3), 2*sqrt(6) and the exact form of E_n(x,y) and F_n(x) enter only through this external result. The manuscript does not compute K_n(e1), K_n(e2), or the relevant central second moment needed to check the assumptions. Without a statement of the corollary and a demonstration that K_n satisfies its hypotheses, the bound in Theorem 2.1 is not established even if the algebra is corrected.

minor comments (3)

[Section 2, before Remark 2.2] The sentence 'for n in N and , y in [0,1]' should read 'for n in N and x,y in [0,1]'.
[Remark 2.2] The notation e1(1-e1) is used after only defining e_i in the Introduction, which is acceptable, but a brief reminder would improve readability.
[Remark 2.3] The phrase 'for y sufficiently close to x' is informal; since the estimate is pointwise and valid for all x,y with x≠y, the statement could be made more precise.

Circularity Check

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No circularity: Theorem 2.1 is a direct application of the author's own general Corollary 4.1 in [4] to Bernstein–Kantorovich operators, not a restatement of it; the cited corollary is a general theorem for positive linear operators, and E_n and F_n are explicit moment expressions rather than fitted or conclusion-defined quantities.

full rationale

The derivation chain is transparent: Corollary 4.1 in [4] is a general semi-discrete quantitative Voronovskaya-type estimate for positive linear operators. The paper defines E_n(x,y) and F_n(x) as concrete expressions obtained from the Bernstein–Kantorovich moments x(1-x)(n-1)/(n+1)^2 + 1/(3(n+1)^2) and (1-2x)/(2(n+1)), substitutes the product fg and the individual f, g into that general corollary, subtracts the f- and g-terms, and isolates the residual |K_n(f)(x)-f(x)| |K_n(g)(x)-g(x)|. The target bound is not an input to the corollary; the constants 1/3, 1/sqrt(3) and 2 sqrt(6) enter as values inherited from the general theorem applied to the second moment of K_n, not by fitting the conclusion to the data or by defining E_n/F_n in terms of the final inequality. Thus the new theorem has independent content as a specialization to Bernstein–Kantorovich polynomials. The proof is terse and does not restate or verify the hypotheses of [4, Cor. 4.1], and it relies on a self-citation that is load-bearing in the sense of supplying the quantitative estimate; but a reliance on a previously proved general theorem, even by the same author, is normal mathematical practice and is not circularity. Nothing in the paper equates two quantities by construction or renames a fitted parameter as a prediction, so no circular step can be exhibited.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper's result is an application of the author's own general theorem; the ledger captures that the main load-bearing assumption is the prior corollary and the key moment identities. No free parameters are fitted and no new entities are introduced.

assumptions (3)

domain assumption Corollary 4.1 in Gal (Result. Math. 2020), a general semi-discrete Voronovskaya-type inequality for positive linear operators.
The proof of Theorem 2.1 is a direct application; the paper does not restate or verify the corollary's hypotheses.
standard math Standard moment identities for Bernstein-Kantorovich polynomials, e.g., K_n(e1)(x)-x = (1-2x)/(2(n+1)) and K_n(e2)(x)-x^2 = O(1/n), used to define E_n and F_n.
The expressions for E_n and F_n in Section 2 are stated without derivation; they are classical but not proven here.
standard math The estimate |K_n(h)(x)-h(x)| <= (1/(2n))||h'|| + (8/(9n))||h''|| from Acu-Gonska [1], used only in Remark 2.2.
Invoked for the asymptotic corollary, not for the main theorem.

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Cite this review

Pith. "Pith review of Semi-discrete Gruss-Voronovskaya-type and Gruss-type estimates for Bernstein-Kantorovich polynomials." pith.science (2026). https://pith.science/paper/Y6XQA7NY

@misc{pith2026201002697,
  author       = {Pith},
  title        = {Pith review of: Semi-discrete Gruss-Voronovskaya-type and Gruss-type estimates for Bernstein-Kantorovich polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y6XQA7NY}},
  note         = {Machine review of arXiv:2010.02697}
}

read the original abstract

The aim of this note is to prove a semi-discrete Gruss-Voronovskaya-type estimate for Bernstein-Kantorovich polynomials. Also, as a consequence, a perturbed Gruss-type estimate is obtained.

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Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[4]

Result Math

Gal, S.G.: Semi-discrete quantitative Voronovskaya-type theor ems for positive linear operators. Result Math. 75(3), Art. 117 (2020)

work page 2020
[1]

Acu, A.M., Gonska, H.: Classical Kantorovich operators revisited . Ukr. Math. J. 71, 843-852 (2019)

work page 2019
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Acu, A.M., Gonska, H., Ra¸ sa, I.: Gr¨ uss-type and Ostrovski-ty pe inequalities in ap- proximation theory. Ukr. Math. J. 63, 843-864 (2011) 4

work page 2011
[3]

Voronovskaya ”D´ etermination de la forme asymptotique de l’approximation des fonctions par les polynˆ omes de M

Bernstein, S.N.: Compl´ ement ` a l’article de E. Voronovskaya ”D´ etermination de la forme asymptotique de l’approximation des fonctions par les polynˆ omes de M. Bern- stein”. C. R. Acad. Sci. URSS 86-92 (1932)

work page 1932
[5]

Gal, S.G., Gonska, H.: Gr¨ uss and Gr¨ uss-Voronovskaya-type e stimates for some Bernstein-type polynomials of real and complex variables. Jaen J. A pprox. 7(1), 97-122 (2015)

work page 2015
[6]

Gavrea, I., Ivan, M.: The Bernstein Voronovskaja-type theor ems for positive linear approximation operators. J. Approx. Theory 192, 291-296 (2015)

work page 2015
[7]

Gonska, H.: On the degree of approximation in Voronovskaya’s th eorem. Stud. Univ. ”Babes-Bolyai” ser. math. LII (3), 103-115 (2007)

work page 2007
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Mathe- matica Slovaca 63(5), 1007-1024 (2013)

Gonska, H., Ra¸ sa, I., Rusu, M.: ˘Ceby˘ sev-Gr¨ uss-type inequalities revisited. Mathe- matica Slovaca 63(5), 1007-1024 (2013)

work page 2013

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[9]

Bernstein, I, II

Kantorovich, L.V.: Sur certains d´ eveloppements suivant les poly nˆ omes de la forme de S. Bernstein, I, II. C.R. Acad. Sci. URSS 563-568, 595-600 (19 30)
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G.: The asymptotic value of the approximation of mu ltiply diﬀeren- tiable functions by positive linear operators

Mamedov, R. G.: The asymptotic value of the approximation of mu ltiply diﬀeren- tiable functions by positive linear operators. Dokl. Akad. Nauk SSSR 146, 1013-1016 (1962)

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[11]

Bernstein

Voronovskaja, E.: D´ etermination de la forme asymptotique de l’approximation des fonctions par les polynˆ omes de M. Bernstein. C. R. Acad. Sci. URSS 79-85 (1932) 5

1932

Pith tools

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

[1] [4]

Result Math

Gal, S.G.: Semi-discrete quantitative Voronovskaya-type theor ems for positive linear operators. Result Math. 75(3), Art. 117 (2020)

work page 2020

[2] [1]

Acu, A.M., Gonska, H.: Classical Kantorovich operators revisited . Ukr. Math. J. 71, 843-852 (2019)

work page 2019

[3] [2]

Acu, A.M., Gonska, H., Ra¸ sa, I.: Gr¨ uss-type and Ostrovski-ty pe inequalities in ap- proximation theory. Ukr. Math. J. 63, 843-864 (2011) 4

work page 2011

[4] [3]

Voronovskaya ”D´ etermination de la forme asymptotique de l’approximation des fonctions par les polynˆ omes de M

Bernstein, S.N.: Compl´ ement ` a l’article de E. Voronovskaya ”D´ etermination de la forme asymptotique de l’approximation des fonctions par les polynˆ omes de M. Bern- stein”. C. R. Acad. Sci. URSS 86-92 (1932)

work page 1932

[5] [5]

Gal, S.G., Gonska, H.: Gr¨ uss and Gr¨ uss-Voronovskaya-type e stimates for some Bernstein-type polynomials of real and complex variables. Jaen J. A pprox. 7(1), 97-122 (2015)

work page 2015

[6] [6]

Gavrea, I., Ivan, M.: The Bernstein Voronovskaja-type theor ems for positive linear approximation operators. J. Approx. Theory 192, 291-296 (2015)

work page 2015

[7] [7]

Gonska, H.: On the degree of approximation in Voronovskaya’s th eorem. Stud. Univ. ”Babes-Bolyai” ser. math. LII (3), 103-115 (2007)

work page 2007

[8] [8]

Mathe- matica Slovaca 63(5), 1007-1024 (2013)

Gonska, H., Ra¸ sa, I., Rusu, M.: ˘Ceby˘ sev-Gr¨ uss-type inequalities revisited. Mathe- matica Slovaca 63(5), 1007-1024 (2013)

work page 2013