Pith. sign in

REVIEW 3 major objections 6 minor 1 cited by

On a novel probabilistic Sampling Kantorovich operators and their application

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Noisy samples can still converge to the true function

desk verdict The main theorem is false as stated: the proof needs ε→0 in L1, which is neither assumed nor implied, and the 'novel' operator is just the classical SK operator applied to noisy inputs. read the letter →

arxiv 2506.12053 v1 pith:KLII7C76 submitted 2025-05-29 math.GM

Digvijay Singh , Rahul Shukla , Karunesh Kumar Singh This is my paper
classification math.GM MSC 41A2541A3546E3047A5847B3894A12
keywords probabilisticsamplingKantorovichoperatorsstochasticuniformintegrabilityL1approximationpeaksignal-to-noiseratiostructuralsimilarityindeximagereconstructionpointwiseconvergence
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

This paper introduces a probabilistic sampling Kantorovich (PSK) operator, built by feeding noisy measurements $f^\beth(x)=f(x)+\varepsilon^\beth(x)$ into the classical sampling operator $S_n$. The central aim is to show that the expected total error $\mathbb{E}[\|P_n^\beth(f)-f\|_{L^1}]$ drops to zero as the sampling density $n$ grows, assuming the noise is bounded in total size and uniformly integrable. If that holds, then averaging corrupted local samples still recovers the original function on average, which is exactly what one wants for reconstruction problems where data arrives with random errors. The paper also compares classical and probabilistic operators on a test function and on a standard test image, reporting PSNR, SSIM, and MAE, and finds the probabilistic version more stable under randomness.

What carries the argument

The central object is the PSK operator $P_n^\beth(f)=S_n(f+\varepsilon^\beth)$, where $S_n$ is the classical sampling Kantorovich operator that averages $f$ over small cells $D_n^a=\prod_j[a_j/n,(a_j+1)/n]$ and convolves with a kernel $\xi$ satisfying unit-sum, boundedness, and decay conditions. The identity that carries the proof is linearity, $P_n^\beth(f)=S_n(f)+S_n(\varepsilon^\beth)$, which separates deterministic approximation from the transported noise. The boundedness lemma $\|S_n(g)\|_{L^1}\le \|\xi\|_{L^1}\|g\|_{L^1}$, valid for all $n$, is what lets the expectation of the noise term be controlled and the final convergence be asserted.

What would settle it

Take $f=0$ and let the noise be a fixed, non-random function, say $\varepsilon^\beth(x)=1$ on $[0,1]$ and $0$ elsewhere. Then the boundedness and uniform-integrability hypotheses hold, yet the classical convergence theorem gives $S_n(\varepsilon^\beth)\to\varepsilon^\beth$ in $L^1$, so the expected error tends to $\|\varepsilon^\beth\|_{L^1}=1$ rather than $0$.

Watch

Extended reading notes

Core claim

Stated in the paper's own terms, the core claim is Theorem 2: for $f\in L^1(\mathbb{R}^q)$ and a noise process $\varepsilon^\beth$ with $\sup_\beth\int_{\mathbb{R}^q}|\varepsilon^\beth(x)|\,dx<\infty$, the PSK operator satisfies $\mathbb{E}[\|P_n^\beth(f)-f\|_{L^1}]\to0$ as $n\to\infty$ under an assumption of uniform integrability. The proof decomposes the operator as $P_n^\beth(f)=S_n(f)+S_n(\varepsilon^\beth)$, applies the classical convergence theorem to $S_n(f)$, and uses the boundedness lemma $\|S_n(g)\|_{L^1}\le L\|g\|_{L^1}$ to handle the transported noise. The theorem's stated hypotheses are boundedness and uniform integrability; the proof's final paragraph notes that concluding the noise term vanishes may need the additional condition $\varepsilon^\beth\to0$ in $L^1$.

Load-bearing premise

The proof relies on the transported noise term shrinking to zero on average, which only follows if the original noise itself is shrinking in size on average, a condition the theorem does not state.

Editorial extensions

If this is right

  • When the hypotheses of Theorem 2 hold, noisy measurements can be used for approximation: increasing the sampling density drives the expected total reconstruction error to zero.
  • The expected error is controlled by the classical approximation error plus the expected noise level, so the rate at which noise decays in $L^1$ determines how fast the probabilistic reconstruction becomes accurate.
  • The operator is uniformly stable under sampling: $\mathbb{E}[\|P_n^\beth(f)\|_{L^1}]$ stays bounded by a constant times the sum of the signal size and the expected noise size, independently of $n$.
  • In image experiments, the paper's reported metrics indicate that PSK operators keep expected PSNR and SSIM relatively stable across window sizes, while classical operators have better metrics only in ideal noiseless conditions.

Reading between the lines

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

  • The paper's Theorem 2 as written is missing an explicit noise-decay assumption: if $\varepsilon^\beth$ is fixed and nonzero, the expected error cannot converge to zero, so a corrected statement should either require $\varepsilon^\beth\to0$ in $L^1$ or aim for the noisy signal $f+\varepsilon^\beth$ as the target.
  • Because the bound is linear, one could lift the qualitative theorem to a rate estimate: if the classical operator converges at a known rate and $\mathbb{E}[\|\varepsilon^\beth\|_{L^1}]$ decays at a known rate, the same proof gives a rate for $\mathbb{E}[\|P_n^\beth(f)-f\|_{L^1}]$.
  • The numerical section compares expected values of nonlinear image metrics; an untested extension would be to check whether optimizing the kernel or window size under expected SSIM, rather than classical SSIM, changes the reconstructed image.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper introduces 'probabilistic Sampling Kantorovich operators' (PSK operators), defined by applying classical sampling Kantorovich operators to noisy measurements f^ℶ(x) = f(x) + ε^ℶ(x), where ε^ℶ is a random perturbation. The main theoretical result is Theorem 2 in Section 3.1, which asserts that, under a uniform L1-boundedness condition on the noise and an unspecified uniform integrability assumption, E‖P_n^ℶ(f) − f‖_{L1} → 0 as n → ∞. The paper also presents numerical experiments on the function e^{-x^2} and the cameraman image, comparing classical and probabilistic operators via L1 error, expected PSNR, SSIM, and MAE.

Significance. If Theorem 2 were correct, the paper would provide a stochastic analogue of the classical convergence theorem for sampling Kantorovich operators, potentially of interest for signal and image processing under noisy measurements. The paper also gives a comparison of classical and probabilistic operators in image quality metrics, which could be useful in practice. However, the central convergence claim is false as stated, and therefore the theoretical contribution collapses. The numerical experiments, while illustrating the behavior of the operators, cannot compensate for the flawed theorem. The paper does not provide machine-checked proofs or reproducible code, and its formal framework is too imprecise to support the claimed novelty.

major comments (3)
  1. [Section 3.1, Theorem 2] Theorem 2 is false as stated. By linearity, P_n^ℶ(f) = S_n(f) + S_n(ε^ℶ). Since ε^ℶ ∈ L1, the classical convergence theorem for SK operators gives S_n(ε^ℶ) → ε^ℶ in L1, so E‖S_n(ε^ℶ)‖_{L1} → E‖ε^ℶ‖_{L1}, which is not zero unless ε^ℶ = 0 in L1. The proof explicitly acknowledges this by saying 'possibly additional assumptions (particularly ε^ℶ → 0 in L1)' are needed; that condition is absent from the theorem's hypotheses and is effectively equivalent to the desired conclusion. Thus the stated assumptions do not imply the claimed convergence.
  2. [Section 4, Example 1, Table 1] The numerical results are consistent with the flaw in Theorem 2. With fixed Gaussian noise of standard deviation 0.02, the probabilistic L1 errors are 0.091, 0.048, 0.027, and 0.018 for n = 5, 15, 25, and 35, respectively. Rather than converging to zero, the errors appear to level off near the noise magnitude, which is exactly what one would expect when S_n(ε^ℶ) → ε^ℶ and ε^ℶ is a nonzero fixed noise field. The table therefore does not provide empirical support for the theorem.
  3. [Section 3.1, Lemma 2] Lemma 2 is a boundedness result and is essentially correct, but its proof uses the assumption sup_ℶ ∫|ε^ℶ| < ∞, where ℶ denotes the sample space. This notation is ambiguous: ε^ℶ is a random field, so the supremum should be taken over realizations or over the probability space. More importantly, Lemma 2 only gives a bound in terms of E‖ε^ℶ‖_{L1}; it does not provide any mechanism for that term to vanish, and no condition in the paper ensures E‖ε^ℶ‖_{L1} → 0. Thus the lemma cannot serve as a step toward Theorem 2 as stated.
minor comments (6)
  1. [Section 3.1, Lemma 1 proof] The substitution step contains a typo: after setting w = nx − k, the displayed computation reads '1/nw ∥ξ∥L1'; it should be n^{-q}∥ξ∥_{L1}. The subsequent conclusion is correct, but the notation is garbled.
  2. [Throughout] The sampling cells are denoted inconsistently as D_n^a, R_n^k, and R_n^a. These notations should be unified and defined precisely in one place.
  3. [Section 3.1, Theorem 2] The theorem statement uses R^d in the supremum condition while the rest of the paper works on R^q; these should be consistent.
  4. [Algorithm 1 and Example 1] The noise level is not specified consistently: Algorithm 1 uses a Gaussian with std = 0.02, while Example 1 writes ε_k ∼ N(0, τ²) without assigning a value to τ. This makes the numerical experiments hard to reproduce.
  5. [Table 3] The last column header 'Var(|Sn(f) − f|)' in the table for PSK operators is misleading; it should refer to the variance of the error for the probabilistic operator P_n^ℶ(f), not for the classical Sn(f).
  6. [Overall presentation] There are numerous typographical and grammatical errors, including 'cordiality' for 'cardinality', 'Preciously' for 'Precisely', and 'receptively' for 'respectively'. A thorough editing pass is needed.

Circularity Check

1 steps flagged · score 6.0 of 10

Theorem 2's proof assumes the noise vanishes in L1, making the claimed convergence equivalent to an unstated assumption.

  1. self definitional [Section 3.1, Theorem 2 proof]
    "Eventually, by the assumption of uniform integrability and possibly additional assumptions (particularly εℶ → 0 in L1), we can conclude: lim n→∞ E [∥Pn(εℶ)∥L1] = 0."

    The proof splits the error as E∥P_n^ℶ(f)−f∥ ≤ ∥S_n(f)−f∥ + E∥S_n(ε^ℶ)∥. The first term vanishes by the classical SK theorem. For the second term, the classical theorem gives S_n(ε^ℶ) → ε^ℶ in L1, so E∥S_n(ε^ℶ)∥ → E∥ε^ℶ∥. Thus the desired limit E∥S_n(ε^ℶ)∥ → 0 is equivalent to ε^ℶ → 0 in L1. The proof inserts exactly that condition as a parenthetical 'additional assumption,' but it is absent from Theorem 2's hypotheses (which only assume sup_ℶ ∫|ε^ℶ| < ∞ and uniform integrability, neither of which forces the L1 norm to zero). Consequently the proof is assuming the very convergence it claims to establish, rather than deriving it from the stated hypotheses.

full rationale

The central theoretical claim is Theorem 2, and its proof is circular at the final step: the noise term E∥S_n(ε^ℶ)∥ is asserted to tend to 0 by 'possibly additional assumptions (particularly ε^ℶ → 0 in L1)', but that condition is exactly what the theorem's conclusion requires in the case f=0, and it is not implied by the stated hypotheses of uniform integrability and bounded L1 norm. This is an internal admission in the paper itself, not an external critique. The earlier lemmas (Lemma 1, Lemma 2) are independent and not circular; they are boundedness estimates. The numerical and image-processing sections are empirical comparisons and do not depend on Theorem 2's validity, so they do not introduce additional circularity. However, because the proof of the main theorem reduces, by construction, to assuming ε^ℶ → 0 in L1, the central theoretical derivation is significantly circular. A score of 6 reflects one load-bearing step that reduces the claimed convergence to an unstated form of the conclusion, while the rest of the paper retains independent empirical content.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central theorem rests on the classical L1 convergence theorem for SK operators and on a hidden assumption that the noise tends to zero in L1. The only hand-chosen numerical parameter is the Gaussian noise standard deviation used in examples. No new physical entities are introduced.

free parameters (1)
  • Gaussian noise standard deviation = 0.02 (variance 0.0004)
    Chosen by hand in Algorithm 1 and Section 4, Example 1; the numerical errors depend on it. It is not fitted to data, but it is an ad hoc input to the experiments.
assumptions (4)
  • standard math Classical SK operators satisfy ||S_n(f)-f||_{L1}->0 for f in L1 (Bardaro et al. [17]).
    Invoked in Theorem 2's proof and accepted from prior literature without reproof.
  • domain assumption The kernel ξ satisfies unit summation, boundedness, and the decay estimate |ξ(x)| ≤ L(1+||x||)^{-q-δ}.
    Assumed in Section 2.1.1 for all results; standard in sampling theory.
  • domain assumption The noise process ε^ℶ is uniformly integrable and sup_ℶ ||ε^ℶ||_{L1} < ∞.
    Stated in the hypotheses of Theorem 2; the paper does not give a precise definition of the uniform integrability used.
  • ad hoc to paper ε^ℶ → 0 in L1 (hidden vanishing-noise assumption).
    Needed for the last step of Theorem 2's proof but absent from the theorem statement; this is effectively the result being proved.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On a novel probabilistic Sampling Kantorovich operators and their application." pith.science (2026). https://pith.science/paper/KLII7C76

@misc{pith2026250612053,
  author       = {Pith},
  title        = {Pith review of: On a novel probabilistic Sampling Kantorovich operators and their application},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KLII7C76}},
  note         = {Machine review of arXiv:2506.12053}
}
read the original abstract

This article starts with the fundamental theory of stochastic type convergence and the significance of uniform integrability in the context of expectation value. A novel probabilistic sampling kantorovich (PSK-operators) is established with the help of classical sampling operators (SK-operators). We establish the proof of the fundamental theorem of approximation and a lemma corresponding to the PSK- operators. Moreover, some examples are illustrated not only in numerical form but also in a detailed study of some important features of an image at different samples. Eventually, a comparative analysis is made on the basis of some parameters like peak signal noise ratio (PSNR), structural similarity index (SSIM) etc. between the classical and probabilistic sense in tabulated form, which connects the whole dots of the theory present in the article.

Figures

Figures reproduced from arXiv: 2506.12053 by the authors.

Figure 1
Figure 1. Approximation of f(x) = e −x 2 using classical SK-Operator for various n. where particularity, εk is a random variable of real value as εk ∼ N (0, τ 2 ) representing noise added to the local average over each interval Ik. Now, we give the numerical data in the following table, where we compare the classical and probabilistic SK- operators at sample level as; Sampling Level n Classical SK Error Probabilistic SK Error… view at source ↗
Figure 2
Figure 2. Approximation of f(x) = e −x 2 using PSK Operators for differ￾ent n. In the real-life situation problems, we construct the behavior of randomness or noisy image (non-ideal situation) for both the operators. In the following problem, let us do a detailed analysis; Example 2. The operators in classical and stochastic operators defined in equations (1.1) and (1.2), respectively, and a discrete function f in the form of… view at source ↗
Figure 3
Figure 3. SK-operators • Mathematical parameters with respect to the PSK-operators [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: SK-operators in probabilistic way Window Size E[PSNR] E[SSIM] E[MAE] Var(|Sn(f) − f|) 3 × 3 28.48 0.8421 0.0305 0.000129 7 × 7 28.98 0.8797 0.0274 0.000081 15 × 15 27.86 0.8446 0.0283 0.000047 [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A comparative study of some wavelet and sampling operators on various features of an image

    cs.CV 2025-07 unverdicted novelty 4.0 of 10

    Sampling Kantorovich-type operators are compared on image metrics, with each operator showing strengths on different features.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages · cited by 1 Pith paper

  1. [1]

    Agrawal, R

    P.N. Agrawal, R. Shukla, and B. Baxhaku, Characterization of deferred type statistical convergence and P-summability method for operators: Applications to q-Lagrange–Hermite operator , Mathemat- ical Methods in the Applied Sciences, 46, 4, 2023, 4449-4465

    work page 2023

  2. [2]

    P. N. Agrawal, B. Baxhaku, & A. Berisha, Enhanced function approximation and applications to image scaling: A new family of exponential sampling neural network Kantorovich operators Appl. Soft Comput., 174, 2025, 112923

    work page 2025

  3. [3]

    R. G. Antonini, T. C. Hu, Y. Kozachenko, & A. Volodin, An application of φ-subgaussian technique to Fourier analysis , Journal of Mathematical Analysis and Applications , 408(1), 2013, 114–124

    work page 2013

  4. [4]

    Bardaro, L

    C. Bardaro, L. Faina, I. Mantellini Quantitative Voronovskaja formulae for multivariate Durrmeyer sampling type series Math. Nachr., 289(14-15), (2016), :1702–1720

    work page 2016

  5. [5]

    Bardaro and I

    C. Bardaro and I. Mantellini, Asymptotic expansion of multivariate Durrmeyer sampling type series , Jaen Journal on Approximation, 6(2), (2014), 143-165

    work page 2014

  6. [6]

    Balser, & M

    W. Balser, & M. Miyake, Analysis-summability of formal solutions of certain partial differential equations Acta Scientiarum Mathematicarum, 65(3-4), (1999), 543-552

    work page 1999

  7. [7]

    W.Balser, Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations ,Universitext, Springer-Verlag, NewYork, 2000

    work page 2000

  8. [8]

    N. L. Braha, Some properties of new modified Sz´ asz-Mirakyan operators in polynomial weight spaces via power summability method Bull. Math. Anal. Appl, 10(3), 2018, 53-65

    work page 2018

Show all 28 references
  1. [9]

    P. L. Butzer and L. S. Rudolf Sampling theory for not necessarily band-limited functions: a historical overview, SIAM review 34.1 (1992): 40-53

    1992

  2. [10]

    Costarelli and G

    D. Costarelli and G. Vinti, Inverse results of approximation and the saturation order for the sampling Kantorovich series, J. Approx. Theory, 242, (2019), 64-82

    2019

  3. [11]

    Vinti, An inverse result of approximation by sampling Kantorovich series , Proc

    D.Costarelli and G. Vinti, An inverse result of approximation by sampling Kantorovich series , Proc. Edinb. Math. Soc., 62(1), (2019), 265-280

    2019

  4. [12]

    Costarelli and G

    D. Costarelli and G. Vinti, Approximation by multivariate multivariate SK-operators in the setting of Orlicz spaces Bollettino dell’Unione Matematica Italiana, 4(3), (2011), 445-468

    2011

  5. [13]

    Costarelli and G

    D. Costarelli and G. Vinti, Saturation by the Fourier transform method for the sampling Kantorovich series based on bandlimited kernels ,Anal. Math. Phys., 9(4), (2019), 2263-2280

    2019

  6. [14]

    M. O. Cabrera, A. N. D. R. E. W. Rosalsky, M. E. H. M. E. T. ¨Unver, & A. Volodin, A new version of uniform integrability via power series summability methods , Theory of Probability and Its Applications, 67(1), 2022, 89–104

    2022

  7. [15]

    D., Gadjiev, & C

    A. D., Gadjiev, & C. Orhan, Some approximation theorems via statistical convergence The Rocky Mountain Journal of Mathematics, 2002, 129-138. 16

    2002

  8. [16]

    M. O. Cabrera, and A. I. Volodin. Mean convergence theorems and weak laws of large numbers for weighted sums of random variables under a condition of weighted integrability Journal of Mathemat- ical Analysis and Applications, 305, 2, 2005, 644-658

    2005

  9. [17]

    Bardaro, G

    C. Bardaro, G. Vinti, P. L. Butzer, & R. L. Stens, Kantorovich-type generalized sampling series in the setting of Orlicz spaces , Sampl. Theory Signal Process. Data Anal. , 6, 2007, 29-52

    2007

  10. [18]

    Demirci, S

    K. Demirci, S. Yıldız, & F. Dirik, Statistical convergence with respect to power series method on time scales. Filomat, 38(13), 2024, 4775-4789

    2024

  11. [19]

    Yurdakadim, Approximation properties of convolution operators via statistical conver- gence based on a power series , Commun

    R.Dinar, & T. Yurdakadim, Approximation properties of convolution operators via statistical conver- gence based on a power series , Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 74(1), 2025, 92-102

    2025

  12. [20]

    T. C. Hu, M. O. Cabrera, & A. I. Volodin, Convergence of randomly weighted sums of Banach space valued random elements and uniform integrability concerning the random weights , Statistics & Probability Letters, 51(2), 2001, 155–164

    2001

  13. [21]

    Robinson, On functional transformations and summability , Proceedings of the London Mathe- matical Society 2.1, 1950, 132-160

    A. Robinson, On functional transformations and summability , Proceedings of the London Mathe- matical Society 2.1, 1950, 132-160

    1950

  14. [22]

    Shukla, P

    R. Shukla, P. N. Agrawal and B. Baxhaku P-summability method applied to multivariate (p, q)- Lagrange polynomial operators, Analysis and Mathematical Physics 12, 6, 2022, 148

    2022

  15. [23]

    Singh and K

    D. Singh and K. K. Singh, Fuzzy approximation theorems via power series summability methods in two variables , Soft Computing 28, 2, 2024, 945-953

    2024

  16. [24]

    S¨ oylemez, & M

    D. S¨ oylemez, & M. ¨Unver, Korovkin type theorems for Cheney–Sharma operators via summability methods Results in Mathematics, 72, (2017), 1601-1612

    2017

  17. [25]

    P. V. Subrahmanyam, Ces` aro summability of fuzzy real numbers, Journal of Analysis, 7, 1999, 159- 168

    1999

  18. [26]

    Tas and T

    E. Tas and T. Yurdakadim, Approximation by positive linear operators inmodular spa ces by power seriesmethod, Positivity, 21, 2017, pp. 1293–1306, https://doi.org/10.1007/s11117-017-0467-z

    2017 doi

  19. [27]

    ¨Unver, Abel transforms of positive linear operators In 11th International Conference of Numerical Analysis and Applied Mathematics 2013: ICNAAM 2013, (Vol

    M. ¨Unver, Abel transforms of positive linear operators In 11th International Conference of Numerical Analysis and Applied Mathematics 2013: ICNAAM 2013, (Vol. 1558, pp., (2013, October). 1148- 1151)

    2013

  20. [28]

    Wang, A.C

    Z. Wang, A.C. Bovik, H. R. Sheikh, & E. P. Simoncelli, Image quality assessment: from error visibility to structural similarity , IEEE Trans. Image Process., 13(4), 2004, 600-612

    2004

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.