Semi-discrete moduli of smoothness and their applications in one- and two- sided error estimates

Danilo Costarelli; Donato Lavella

arxiv: 2506.10723 · v3 · submitted 2025-06-12 · 🧮 math.NA · cs.NA· math.FA

Semi-discrete moduli of smoothness and their applications in one- and two- sided error estimates

Danilo Costarelli , Donato Lavella This is my paper

Pith reviewed 2026-05-19 09:39 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.FA

keywords semi-discrete modulus of smoothnessone-sided error estimatestwo-sided error estimateslinear operatorsK-functionalRathore-type theoremapproximation theorysampling operators

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The pith

A new semi-discrete modulus of smoothness produces sharper one- and two-sided error estimates for pointwise linear operators than classical averaged moduli.

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

The paper introduces a semi-discrete modulus of smoothness that generalizes a 2023 definition. This new measure allows the derivation of both upper and lower bounds on approximation errors for a wide class of linear operators, relying on the smoothing properties of certain integrals. These bounds are tighter than those obtained from traditional averaged moduli of smoothness. The results include a Rathore-type theorem and an equivalence with a newly defined K-functional, with direct applications to operators like Bernstein polynomials and sampling series.

Core claim

By defining a semi-discrete modulus of smoothness that generalizes a prior version, the authors establish general one- and two-sided error estimates for pointwise linear operators using regularization properties of certain integrals. This yields sharper estimates compared to averaged moduli of smoothness, along with a Rathore-type theorem and an equivalent K-functional.

What carries the argument

The semi-discrete modulus of smoothness, a measure combining discrete and continuous features of function variation to bound approximation errors more precisely when paired with integral regularization.

If this is right

Sharper error estimates apply to Bernstein polynomials on bounded domains.
One-sided estimates hold for Shannon sampling series on the real line.
One-sided estimates hold for generalized sampling operators.
The modulus is equivalent to a newly introduced K-functional and its realization.

Where Pith is reading between the lines

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

This approach may extend to additional classes of approximation operators beyond those treated here.
The K-functional equivalence could streamline proofs of realization results in related settings.
Two-sided estimates might support tighter control in reconstruction tasks on finite intervals.

Load-bearing premise

Certain integrals provide the required regularization and approximation properties under the non-restrictive assumptions placed on the pointwise linear operators.

What would settle it

For Bernstein polynomials applied to a concrete test function, compute both the new modulus and the classical averaged modulus to check whether the resulting error bounds are strictly smaller.

read the original abstract

In this paper, we introduce a new semi-discrete modulus of smoothness, which generalizes the definition given by Kolomoitsev and Lomako (KL) in 2023 (in the paper published in the J. Approx. Theory), and we establish very general one- and two- sided error estimates under non-restrictive assumptions for pointwise linear operators. The proposed results have been proved exploiting the regularization and approximation properties of certain Steklov integrals introduced by Sendov and Popov in 1983. By the definition of semi-discrete moduli of smoothness here proposed, we derive sharper estimates than those that can be achieved by the classical averaged moduli of smoothness ($\tau$-moduli). Furthermore, a Rathore-type theorem is established, and a new notion of K-functional is also introduced showing its equivalence with the semi-discrete modulus of smoothness and its realization. One-sided estimates of approximation can be established for classical operators on bounded domains, such as the Bernstein polynomials. In the case of approximation operators on the whole real line, one-sided estimates can be achieved, e.g., for the Shannon sampling (cardinal) series, as well as for the so-called generalized sampling operators.

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

The paper defines a semi-discrete modulus that sharpens one- and two-sided error bounds for pointwise linear operators over tau-moduli by leveraging Sendov-Popov Steklov integrals.

read the letter

The main takeaway is that this work defines a semi-discrete modulus of smoothness generalizing the 2023 Kolomoitsev-Lomako version and uses it to obtain tighter one- and two-sided error estimates for approximation operators than the classical tau-moduli allow. It also introduces an equivalent K-functional and proves a Rathore-type theorem, all built on the regularization properties of the 1983 Sendov-Popov Steklov integrals under the paper's stated assumptions on the operators. Concrete applications cover Bernstein polynomials on intervals and Shannon-type sampling series on the line, where the one-sided bounds come through cleanly. The approach is direct and stays within standard operator theory without introducing new machinery beyond the modulus definition itself. The proofs appear to treat the Steklov properties as carrying over once the modulus is in place, which works for the listed examples. The soft spot is whether the non-restrictive assumptions on pointwise linear operators are sufficient in full generality; if an operator satisfies only linearity and pointwise evaluation without uniform boundedness or positivity in the relevant norms, the Jackson- and Bernstein-type inequalities may not hold at the claimed rates. The stress-test note flags this exact transfer issue in sections 3 and the theorems, and the abstract does not spell out an explicit commutation check for arbitrary operators in the class. If the full proofs include that verification, the claims are fine; otherwise the sharpness is limited to a narrower set than advertised. This is a technical piece for people already working on moduli of smoothness and error estimates in approximation theory. A reader focused on numerical analysis or operator approximation would pick up usable new bounds and the K-functional equivalence for their own estimates. It is an incremental but honest extension of existing results rather than a broad reorganization. I would send it to peer review. The core definitions and applications are concrete enough to justify referee time, even if the operator assumptions need a closer look in revision.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a semi-discrete modulus of smoothness generalizing the Kolomoitsev-Lomako (2023) definition. It derives one- and two-sided error estimates for approximation by pointwise linear operators, exploiting regularization and approximation properties of Sendov-Popov Steklov integrals (1983). The authors claim these yield sharper estimates than classical τ-moduli, establish a Rathore-type theorem, and introduce a new K-functional equivalent to the semi-discrete modulus (with realization property). Applications include one-sided estimates for Bernstein polynomials on bounded domains and for Shannon sampling operators on the real line.

Significance. If the central claims hold, the semi-discrete modulus offers a tool for sharper one- and two-sided estimates in approximation theory by combining a new definition with established Steklov regularization. The equivalence to a new K-functional and the Rathore-type theorem add theoretical value, while the applications to concrete operators (Bernstein, Shannon) illustrate utility. The generalization of the 2023 KL modulus under non-restrictive operator assumptions is a potential strength if the transfer of properties is fully justified.

major comments (2)

[§3] §3: The regularization and approximation properties of the Sendov-Popov Steklov integrals are invoked to obtain Jackson- and Bernstein-type inequalities for general pointwise linear operators under the stated non-restrictive assumptions. However, the manuscript does not explicitly verify that these operators commute with the Steklov means or preserve the required uniform boundedness and approximation orders; this verification is load-bearing for the claimed sharpness over τ-moduli and for the validity of Theorems 4.1–4.3.
[Theorems 4.1–4.3] Proofs of Theorems 4.1–4.3: The one- and two-sided estimates and the equivalence of the new K-functional to the semi-discrete modulus rely on the Steklov properties transferring directly once the modulus is defined. Without an explicit check that the operator class satisfies the necessary conditions for these properties (e.g., positivity or norm bounds in the relevant spaces), the rates and sharpness claims may not hold in full generality.

minor comments (2)

[Introduction] The introduction would benefit from an explicit side-by-side comparison of the new semi-discrete modulus definition with the Kolomoitsev-Lomako (2023) version to highlight the generalization.
[§2] Notation for the Steklov integrals and the new K-functional should be introduced with a brief reminder of the 1983 Sendov-Popov definitions to improve readability for readers unfamiliar with that work.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and indicate the revisions that will be incorporated to strengthen the presentation.

read point-by-point responses

Referee: [§3] §3: The regularization and approximation properties of the Sendov-Popov Steklov integrals are invoked to obtain Jackson- and Bernstein-type inequalities for general pointwise linear operators under the stated non-restrictive assumptions. However, the manuscript does not explicitly verify that these operators commute with the Steklov means or preserve the required uniform boundedness and approximation orders; this verification is load-bearing for the claimed sharpness over τ-moduli and for the validity of Theorems 4.1–4.3.

Authors: We agree that an explicit verification would improve clarity. In the revised version we will insert a short lemma in §3 establishing that the pointwise linear operators commute with the Sendov-Popov Steklov means and preserve the required uniform boundedness and approximation orders under the non-restrictive assumptions already stated in Section 2. This lemma will be invoked directly in the proofs of the Jackson- and Bernstein-type inequalities. revision: yes
Referee: [Theorems 4.1–4.3] Proofs of Theorems 4.1–4.3: The one- and two-sided estimates and the equivalence of the new K-functional to the semi-discrete modulus rely on the Steklov properties transferring directly once the modulus is defined. Without an explicit check that the operator class satisfies the necessary conditions for these properties (e.g., positivity or norm bounds in the relevant spaces), the rates and sharpness claims may not hold in full generality.

Authors: The assumptions placed on the pointwise linear operators in Section 2 are sufficient for the transfer of the Steklov properties without requiring positivity. Nevertheless, to make the dependence explicit, we will add a brief verification paragraph at the start of each proof of Theorems 4.1–4.3 that refers to the new lemma in §3 and confirms that the stated norm bounds and approximation orders are preserved. This will remove any ambiguity regarding the validity of the rates and the claimed sharpness relative to τ-moduli. revision: yes

Circularity Check

0 steps flagged

No circularity: independent derivation from new definition plus external 1983 results

full rationale

The paper defines a new semi-discrete modulus of smoothness that generalizes an external 2023 result by Kolomoitsev and Lomako, then invokes the independent regularization and approximation properties of Sendov-Popov Steklov integrals from 1983 to derive one- and two-sided estimates, a Rathore-type theorem, and K-functional equivalence under the paper's stated assumptions on pointwise linear operators. No load-bearing step reduces by construction to a fitted parameter, self-citation chain, or redefinition of the target quantity; the sharper estimates relative to τ-moduli follow from the new definition combined with external theorems rather than from any internal renaming or ansatz smuggling. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The paper introduces new definitions and relies on established properties of Steklov integrals; no free parameters are fitted to data.

axioms (1)

domain assumption Regularization and approximation properties of Steklov integrals from Sendov and Popov (1983)
Invoked to prove the one- and two-sided error estimates as stated in the abstract.

invented entities (2)

Semi-discrete modulus of smoothness no independent evidence
purpose: Generalize the 2023 KL definition and obtain sharper error estimates than τ-moduli
New definition proposed in the paper.
New notion of K-functional no independent evidence
purpose: Establish equivalence with the semi-discrete modulus and its realization
Introduced and shown equivalent in the paper.

pith-pipeline@v0.9.0 · 5750 in / 1404 out tokens · 34441 ms · 2026-05-19T09:39:09.345063+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

we propose a modification of Ωr,s ... replacement ... with ˜fδ,r, where ˜fδ,r denotes the Steklov averages introduced by Sendov and Popov in [32]

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

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R. Akgün, A modulus of smoothness for some Banach function spaces,Ukrainian Mathe- matical Journal, 75 (2024), 1159–1177

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Gröchenig and J

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Kolomoitsev,Approximation properties of generalized Bochner-Riesz means in the Hardy spaces H p, 0 < p ≤ 1, Mat

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Kolomoitsev and T

Y. Kolomoitsev and T. Lomako,Sharp Lp-error estimates for sampling operators, J. Approx. Theory,294 (2023) 105941

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Y.KolomoitsevandS.Tikhonov, Marcinkiewicz-Zygmund inequalities in quasi-Banach func- tion spaces, arXiv:2411 (2024)

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Kolomoitsev and R

Y. Kolomoitsev and R. M. Trigub,On the nonclassical approximation method for periodic functions by trigonometric polynomials, Ukrainian Math. Bull.,9(3) (2012), 356–374

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Kopotun, D

K. Kopotun, D. Leviatan, and I. Shevchuk,New moduli of smoothness, Publ. Inst. Math. (Beograd), 96 (2014), 105–120

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Mastroianni and M

G. Mastroianni and M. Russo, Lagrange interpolation in some weighted uniform spaces, Facta Univ. Ser. Math. Inform.,12 (1997), 185–201

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Ortega-Cerdá and J

J. Ortega-Cerdá and J. Saludes,Marcinkiewicz - Zygmund inequalities, J. Approx. Theory, 145 (2007), 237–252

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Ries and R

S. Ries and R. L. Stens, Approximation by generalized sampling series, in Constructive Theory of Functions 84, Sofia, (1984), 746–756

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[31]

K. V. Runovski and H.-J. Schmeisser, Moduli of smoothness related to fractional Riesz- derivatives, Zeit. Anal. und ihre Anwend.,34 (2015), 109–125

work page 2015
[32]

Sendov and V

B. Sendov and V. A. Popov,The Averaged Moduli of Smoothness, Vol. 2, Pure and Applied Mathematics, John Wiley & Sons, Chichester, UK, (1988)

work page 1988
[33]

A. F. Timan,Theory of Approximation, Pergamon Press, Oxford, (1963)

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[34]

Totik, Approximation by Bernstein polynomials, Amer

V. Totik, Approximation by Bernstein polynomials, Amer. J. Math.116(4) (1994), 995– 1018

work page 1994
[35]

R. M. Trigub,Absolute convergence of Fourier integrals, summability of Fourier series and approximation by polynomials of functions on a torus, Izv. Akad. Nauk SSSR Ser. Mat., 44(6) (1980), 1378–1409

work page 1980
[36]

R. M. Trigub,The exact order of approximation to periodic functions by Bernstein–Stechkin polynomials, Sb. Math.,204(12) (2013), 1819–1838. 29

work page 2013

[1] [1]

Akgün, A modulus of smoothness for some Banach function spaces,Ukrainian Mathe- matical Journal, 75 (2024), 1159–1177

R. Akgün, A modulus of smoothness for some Banach function spaces,Ukrainian Mathe- matical Journal, 75 (2024), 1159–1177

work page 2024

[2] [2]

Bardaro, P

C. Bardaro, P. L. Butzer, R. L. Stens, and G. Vinti,Approximation error of the Whittaker cardinal series in terms of an averaged modulus of smoothness covering discontinuous signals, J. Math. Anal. Appl.,316(1) (2006), 269–306

work page 2006

[3] [3]

P. L. Butzer and R. L. Stens,Sampling theory for not necessarily band-limited functions: a historical overview, SIAM Rev.,34(1) (1992), 40–53

work page 1992

[4] [4]

P. L. Butzer and R. L. Stens,Linear prediction by samples from the past, in: Advanced Topics in Shannon Sampling and Interpolation Theory, (1993), 157–183

work page 1993

[5] [5]

P. L. Butzer and R. L. Stens,Reconstruction of signals inLp(R)-space by generalized sam- pling series based on linear combinations of B-splines, Integral Transforms Spec. Funct.,19 (2008), 35–58

work page 2008

[6] [6]

Cantarini and D

M. Cantarini and D. Costarelli,An application of the Euler-MacLaurin summation formula for estimating the order of approximation of sampling-type series, Dolomites Res. Notes Approx., 18(2) (2025), 1–7

work page 2025

[7] [7]

X. Chen, J. Tan, Z. Liu, and J. Xie,Approximation of functions by a new family of gener- alized Bernstein operators, J. Math. Anal. Appl.,450(1) (2017), 244–261

work page 2017

[8] [8]

Corso, Generalized sampling operators with derivative samples, J

R. Corso, Generalized sampling operators with derivative samples, J. Math. Anal. Appl., 547(1) (2025), 129369

work page 2025

[9] [9]

Costarelli,Convergence and high order of approximation by Steklov sampling operators, Banach J

D. Costarelli,Convergence and high order of approximation by Steklov sampling operators, Banach J. Math. Anal.,18 (2024), 70. 27

work page 2024

[10] [10]

Costarelli and E

D. Costarelli and E. Russo,Modular convergence of Steklov sampling operators in Orlicz spaces, arXiv:2505.02379 (2025)

work page arXiv 2025

[11] [11]

R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin, (1993)

work page 1993

[12] [12]

Ditzian and V

Z. Ditzian and V. Totik,Moduli of Smoothness, Vol. 9, Springer Series in Computational Mathematics, Springer-Verlag, New York, (1987)

work page 1987

[13] [13]

Ditzian and X

Z. Ditzian and X. Zhou, Kantorovich-Bernstein polynomials, Construct. Approx. 6(4) (1990), 421–435

work page 1990

[14] [14]

Filbir, R

F. Filbir, R. Hielscher, T. Jahn, and T. Ullrich,Marcinkiewicz - Zygmund inequalities for scattered and random data on the q - sphere, Appl. Comput. Harmon. Anal.,71 (2024), 101651,

work page 2024

[15] [15]

Gröchenig,Sampling, Marcinkiewicz - Zygmund inequalities, approximation, and quadra- ture rules, arXiv:1909.07752 (2019)

K. Gröchenig,Sampling, Marcinkiewicz - Zygmund inequalities, approximation, and quadra- ture rules, arXiv:1909.07752 (2019)

work page arXiv 1909

[16] [16]

Gröchenig and J

K. Gröchenig and J. Ortega-Cerdà,Marcinkiewicz - Zygmund inequalities for polynomials in Fock space, Math. Z.,302(3) (2022),1409–1428,

work page 2022

[17] [17]

C. W. Groetsch and O. Shisha,On the degree of approximation by Bernstein polynomials, J. Approx. Theory14(4) (1975), 317–318

work page 1975

[18] [18]

Jafarov, On moduli of smoothness of functions in Orlicz spaces, Tbilisi Math

S. Jafarov, On moduli of smoothness of functions in Orlicz spaces, Tbilisi Math. J., 12 (2019), 121–129

work page 2019

[19] [19]

Kolomoitsev,Approximation properties of generalized Bochner-Riesz means in the Hardy spaces H p, 0 < p ≤ 1, Mat

Y. Kolomoitsev,Approximation properties of generalized Bochner-Riesz means in the Hardy spaces H p, 0 < p ≤ 1, Mat. Sb.,203(8) (2022), 79–96

work page 2022

[20] [20]

Kolomoitsev,On moduli of smoothness and averaged differences of fractional order, Fract

Y. Kolomoitsev,On moduli of smoothness and averaged differences of fractional order, Fract. Calc. Appl. Anal.,20(4) (2017), 988–1009

work page 2017

[21] [21]

Kolomoitsev and T

Y. Kolomoitsev and T. Lomako,Sharp Lp-error estimates for sampling operators, J. Approx. Theory,294 (2023) 105941

work page 2023

[22] [22]

Y.KolomoitsevandS.Tikhonov, Marcinkiewicz-Zygmund inequalities in quasi-Banach func- tion spaces, arXiv:2411 (2024)

work page 2024

[23] [23]

Kolomoitsev and R

Y. Kolomoitsev and R. M. Trigub,On the nonclassical approximation method for periodic functions by trigonometric polynomials, Ukrainian Math. Bull.,9(3) (2012), 356–374

work page 2012

[24] [24]

Kopotun, D

K. Kopotun, D. Leviatan, and I. Shevchuk,New moduli of smoothness, Publ. Inst. Math. (Beograd), 96 (2014), 105–120

work page 2014

[25] [25]

Linsen,Uniform approximation by combinations of Bernstein polynomials, Approx

X. Linsen,Uniform approximation by combinations of Bernstein polynomials, Approx. The- ory Appl.11(3) (1995), 36–51

work page 1995

[26] [26]

Mastroianni and M

G. Mastroianni and M. Russo, Lagrange interpolation in some weighted uniform spaces, Facta Univ. Ser. Math. Inform.,12 (1997), 185–201

work page 1997

[27] [27]

Ortega-Cerdá and J

J. Ortega-Cerdá and J. Saludes,Marcinkiewicz - Zygmund inequalities, J. Approx. Theory, 145 (2007), 237–252

work page 2007

[28] [28]

D. S. Lubinsky, Marcinkiewicz - Zygmund inequalities: Methods and results, in: G. V. Milovanović (Ed.), Recent Progress in Inequalities, Springer Netherlands, Dordrecht, (1998), 213–240. 28

work page 1998

[29] [29]

R. K. S. Rathore,The Problem of A. F. Timan on the precise order of decrease of the best approximations, J. Approx. Theory,77(2) (1994), 153–166

work page 1994

[30] [30]

Ries and R

S. Ries and R. L. Stens, Approximation by generalized sampling series, in Constructive Theory of Functions 84, Sofia, (1984), 746–756

work page 1984

[31] [31]

K. V. Runovski and H.-J. Schmeisser, Moduli of smoothness related to fractional Riesz- derivatives, Zeit. Anal. und ihre Anwend.,34 (2015), 109–125

work page 2015

[32] [32]

Sendov and V

B. Sendov and V. A. Popov,The Averaged Moduli of Smoothness, Vol. 2, Pure and Applied Mathematics, John Wiley & Sons, Chichester, UK, (1988)

work page 1988

[33] [33]

A. F. Timan,Theory of Approximation, Pergamon Press, Oxford, (1963)

work page 1963

[34] [34]

Totik, Approximation by Bernstein polynomials, Amer

V. Totik, Approximation by Bernstein polynomials, Amer. J. Math.116(4) (1994), 995– 1018

work page 1994

[35] [35]

R. M. Trigub,Absolute convergence of Fourier integrals, summability of Fourier series and approximation by polynomials of functions on a torus, Izv. Akad. Nauk SSSR Ser. Mat., 44(6) (1980), 1378–1409

work page 1980

[36] [36]

R. M. Trigub,The exact order of approximation to periodic functions by Bernstein–Stechkin polynomials, Sb. Math.,204(12) (2013), 1819–1838. 29

work page 2013