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On wavelet-based sampling Kantorovich operators and their study in multi-resolution analysis

T0 review · 5 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that a wavelet-based sampling-Kantorovich operator approximates every sufficiently smooth function, with an error of order 2^(-jm) determined by the scale j and the number m of vanishing moments of the kernel.

desk verdict The paper's central approximation theorem fails for the operator as defined, because the missing 2^j normalization is silently inserted during the proof; the useful wavelet-connection motivation does not rescue it. read the letter →

arxiv 2506.18912 v1 pith:25E3CWTP submitted 2025-05-29 math.FA

Digvijay Singh , Rahul Shukla , Karunesh Kumar Singh This is my paper
classification math.FA MSC 41A2541A3546E3047A5847B3894A12
keywords waveletsamplingoperatorsKantorovichtransformconvolutionmultiresolutionanalysisorderofapproximationvanishingmomentsmoduluscontinuity
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 proposes a family of wavelet-based filtering operators, built from a scaling kernel $\kappa_1$ and a wavelet kernel $\chi_2$, where each sample is replaced by a local average against $\chi_2$. The central claim is that these operators converge to the original function as the scale parameter grows, and that the error obeys an explicit bound in terms of the function's derivatives and a modulus of continuity. For sufficiently smooth functions and kernels with $m$ vanishing moments, the paper further claims a rate of convergence of order $2^{-jm}$. The authors use this to argue that the operator approximates well on smooth regions while the error near sharp edges is harder to control, which they illustrate with B-spline and sine examples.

What carries the argument

The central object is the operator itself: a partition-of-unity scaling kernel $\kappa_1$ performs the sampling, while the wavelet kernel $\chi_2$ acts as a local averaging window. For smooth $h$, the proof expands $h$ in a Taylor series around $t$, uses the compact support and nonnegativity of $\kappa_1$ to bound the sums, and controls the remainder with the modulus of continuity of the highest derivative. For Theorem 6, the vanishing moments of $\chi_2$ remove the lower-order Taylor terms, leaving an error term proportional to the $m$-th moment of $\chi_2$ divided by $2^{jm}$. Earlier results on convolution for wavelet transforms, involving the basic function $D(t_1,t_2,t_3)$, place the operator inside the wavelet-transform framework.

What would settle it

Take $h(t)=1$, let $\kappa_1$ be the piecewise-constant scaling kernel, and let $\chi_2$ have zero integral. Substituting into the operator as written, each integral $\int_{\mathbb{R}}\chi_2(2^j u-\varrho)\,du$ is zero, so the operator returns $0$, not $1$, which contradicts convergence to $h$.

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Extended reading notes

Core claim

The central claim is that the operator $(W_{\kappa_1,\chi_2} h)(t) = \sum_{\varrho\in\mathbb{Z}} \kappa_1(2^j t - \varrho) \int_{\mathbb{R}} h(u) \chi_2(2^j u - \varrho)\,du$ defines a convergent approximation scheme for functions $h\in C^N(\mathbb{R})$, provided $\kappa_1$ is a nonnegative bounded scaling function with compact support and unit partition of unity. Theorem 3 states the pointwise error is bounded by $\sum_{i=1}^{\varrho} |h^{(i)}(t)|/i! \, a^i/2^{i\varrho} + a^N/2^{N\varrho}\, \omega_1(h^{(N)}, a/2^j)$, so the error decays as the scale grows. Theorem 6 states that when $\chi_2$ has $m$ vanishing moments and $h\in C^m$ with bounded $m$-th derivative, the error is at most $C 2^{-jm}$, a rate that improves with both scale and the number of vanishing moments. The paper also derives related bounds via the wavelet-transform convolution identity and examines the behaviour of the error near discontinuities.

Load-bearing premise

The whole argument depends on the averaging integrals being normalized so that the operator reproduces constant functions; without that scale factor the operator does not generally converge to the function it is meant to approximate.

Editorial extensions

If this is right

  • If the error bound of Theorem 6 is correct, the operator offers a constructive way to approximate $C^m$ functions with arbitrarily high algebraic rate by choosing kernels with more vanishing moments.
  • The pointwise bound in Theorem 3 implies that the operator converges to $h$ at every point where $h$ and its derivatives are bounded, and that the first term decays exponentially in the scale parameter.
  • The authors' analysis indicates that near discontinuities the guaranteed error bound no longer improves with scale, matching the observed erratic numerical behaviour at such points.
  • The sampling-theorem estimates in Section 3.3 imply that even when the function is not in the multiresolution space, the error between the reconstructed and projected versions can be controlled by a band-limiting measure $\delta$.

Reading between the lines

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

  • A direct numerical check on $h(t)=1$ at a few scales would reveal whether the operator as defined is missing an explicit $2^j$ normalization factor.
  • If the $O(2^{-jm})$ rate holds, using kernels with more vanishing moments would push the error down at an algebraic rate, and in the limit of bandlimited kernels one might recover exponential convergence analogous to classical sampling.
  • The construction via the basic function $D(t_1,t_2,t_3)$ could be generalized: any pair of scaling and wavelet functions satisfying the same convolution identity would produce a sampling-Kantorovich operator with the same error structure.
  • The authors' observation about sharp edges implies a practical use: the local error of the operator could act as an edge detector, since the error stays large and scale-independent near discontinuities while decaying smoothly elsewhere.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 6 minor

Summary. The paper defines a wavelet-based sampling Kantorovich operator (W_{κ1,χ2_ϱ} h)(t) = Σ_{ϱ∈Z} κ1(2^j t − ϱ) ∫_R h(u) χ2(2^j u − ϱ) du in (1.4) and claims that it approximates h as the scale parameter grows. The main results are Theorem 3, which asserts a pointwise error bound in terms of derivatives and a modulus of continuity; Theorem 6, which asserts an exponential-type error bound C2^{−jm} for kernels with m vanishing moments; Theorems 4–5, which state bounds involving wavelet-transform convolution; and a section on sampling-theorem error estimates (Theorem 7 and propositions). Numerical examples in Section 4 are offered as empirical support. The central claim is that the operator converges to the target function with a rate depending on smoothness and scale.

Significance. If the approximation claim were correct, the paper would contribute a wavelet-based analogue of sampling Kantorovich operators with explicit convergence rates, of potential interest in approximation theory and signal processing. However, the central claim is not established: the operator as defined in (1.4) lacks the scale normalization required for convergence, and the proofs of the main theorems introduce unsupported factors and silently drop the kernel χ2. The paper provides no reproducible code or machine-checked derivations, and the numerical example in Table 1 does not show the claimed decay. The stated results therefore do not support the advertised conclusions.

major comments (5)
  1. [Definition (1.4) and Theorem 3] The operator as defined in (1.4) lacks the 2^j scaling factor needed for convergence in a sampling-Kantorovich setting. Taking h ≡ 1 and using the partition-of-unity hypothesis Σ_{ϱ} κ1(t − ϱ) = 1 from Theorem 3, the operator gives W(1)(t) = Σ_{ϱ} κ1(2^j t − ϱ) ∫_R χ2(2^j u − ϱ) du = 2^{−j} (∫_R χ2(v) dv), by the change of variable v = 2^j u − ϱ. This tends to 0 as j → ∞ unless ∫χ2 grows like 2^j, impossible for a fixed kernel; if χ2 has vanishing moments, the value is identically 0. Thus the operator (1.4) cannot approximate even constant functions, and the central convergence claim fails.
  2. [Proof of Theorem 3] The proof does not prove the stated result for the operator (1.4). The first displayed equality inserts a factor 2^{j/2} and a term −2^{−j/2} h(t) inside the sum, neither of which appears in the definition (1.4). The second equality changes variables to v = 2^j u − ϱ but keeps the 2^{−j/2} h(t) term, and the third equality replaces ∫ h(v/2^j) χ2(v−k) dv by ∫ h(v/2^j) dv, silently dropping the kernel χ2. No hypothesis in the paper justifies this substitution. Consequently the bound (3.1) is proved, at best, for a different, implicitly renormalized operator.
  3. [Theorem 6] Theorem 6 assumes that χ2 has m vanishing moments, which for k = 0 gives ∫_R χ2(v) dv = 0 when m ≥ 1. As shown in the first major comment, this forces W(1) = 0 for the operator defined by (1.4), so the operator cannot reproduce constant functions and the error bound (3.2) is not an approximation result for (1.4). Additionally, the proof of Theorem 6 changes the argument of κ1 from 2^j t − ϱ in the definition to 2^j u − ϱ in the display after the Taylor expansion, and defines C2 = Σ κ1(2^j − ϱ), which is not the sum appearing in (1.4).
  4. [Example 3 and Table 1] The numerical table does not show the claimed convergence. For t = 0.1000, the error E(t) is 0.5878 for every ϱ = 1, 2, 3, 4; for t = 0.3000 it is 0.8851 at ϱ = 1 but 0.9511 at ϱ = 2, 3, 4; and at t = 0.5000 it is 0.0000 for ϱ = 0, 1, 3, 4 but 0.0659 for ϱ = 2. No monotone decrease with ϱ is visible, and no computational details or code are provided to reproduce the numbers.
  5. [Theorems 4 and 5] The inequalities in Theorems 4 and 5 are presented as upper bounds for the approximation error, but their right-hand sides contain undefined constants K(p, ρ), K'(p, ρ), G(p, ρ), G'(p, ρ) and do not visibly depend on the scale parameter j or on any parameter that tends to zero. No argument is given to show that these bounds tend to zero, so the results do not establish a rate of convergence. The hypotheses also mix 1/p + 1/q = 1 + ρ with 1/r1 + 1/r2 + ν = 2 without relating the parameters to the error.
minor comments (6)
  1. [Notation throughout] The symbol ϱ is used both as the summation index in (1.4) and as the exponent in 1/p + 1/q = 1 + ϱ in Theorem 1 and Section 3.2, making those statements ambiguous.
  2. [Proposition 1] Proposition 1 refers to 'the sampling theorem stated as in (3.6)', but no equation (3.6) appears before that point; the label (3.6) appears only later inside the proof of Theorem 7.
  3. [Constants in Theorems 1, 4, 5] The constants K(p, ρ), K'(p, ρ), G(p, ρ), and G'(p, ρ) are never defined, so the estimates in these theorems are not quantitative and cannot be checked numerically.
  4. [Example 2] Example 2 reports a numerical interval 0.00866839702279679 ≤ ||·|| ≤ 0.024626088238854097 but does not define the specific kernel χ1, the parameters J and δ used in B1, the function W, or the Python computation, so the result cannot be reproduced.
  5. [Reference list] Reference [31] contains an unofficial sci-hub URL, which is not an acceptable citable source; if the article is to be cited, the standard journal DOI should be used.
  6. [Language and typos] The manuscript contains many typographical and grammatical errors, including 'madulus' for 'modulus', 'Preciously' for 'Precisely', 'prepositions' for 'propositions', and 'interferences' for 'inferences', which hinder readability.

Circularity Check

2 steps flagged · score 6.0 of 10

Theorem 3's convergence is not derived from (1.4): the proof inserts the missing 2^{j/2} normalization and the subtracted h(t) term into the first line, then drops chi2 from the sampling integral, so the central approximation claim is built into the proof rather than obtained from the operator as defined.

  1. self definitional [Section 3.1, Proof of Theorem 3, first displayed equality (after Eq. (3.1))]
    "|(W^{κ1,χ2}_ϱ h)(t) − h(t)| = | 2^{j/2} Σ_{ϱ=−∞}^{+∞} [ ∫_R h(u)χ2(2^j u − ϱ) du − 2^{−j/2} h(t)] κ1(2^j t − ϱ) |"

    Definition (1.4) gives W h = Σ κ1(2^j t−ϱ) ∫ h(u)χ2(2^j u−ϱ)du, with no 2^{j/2} prefactor and no −2^{−j/2}h(t) term inside the sum. The proof's first equality inserts both, and since the theorem assumes Σ κ1(t−ϱ)=1, the subtracted term is exactly the target h(t). Thus the convergence that Theorem 3 is supposed to establish is placed into the expression being bounded; the subsequent estimate applies to a renormalized operator different from (1.4).

  2. other [Section 3.1, Proof of Theorem 3, third displayed equality]
    "= | 2^{j/2} Σ_{ϱ=−∞}^{+∞} [ ∫_R h(v/2^j) dv − h(t)] χ2(v−k) κ1(2^j t−k) |"

    The displayed line before it has ∫ h(v/2^j) χ2(v−k) dv inside the bracket; this line moves χ2(v−k) outside the integral and integrates h(v/2^j) alone. No stated assumption of Theorem 3 (compact support, non-negativity, partition of unity) licenses deleting χ2 from the sampling integral. This deletion is what makes the proof's expression resemble a normalized sampling-Kantorovich expansion, so the bound (3.1) is proved for a different operator in which χ2 has been stripped from the averaging operation.

full rationale

No fitted parameters are present, and the main auxiliary estimates (Theorems 1 and 2 and the Hardy-Littlewood-Sobolev step) are cited from external works, so the paper is not circular by self-citation. The circularity is confined to the core proof: Theorem 3 begins by rewriting W h − h with a 2^{j/2} prefactor and an internal −2^{−j/2}h(t) term, neither of which occurs in definition (1.4); with the partition of unity this term returns exactly h(t), so the convergence target is assumed inside the expression to be bounded. The subsequent deletion of χ2 from the integral further shows that the theorem is proved for an implicitly renormalized, χ2-free sampling expression, not for the operator (1.4). Theorem 6's vanishing-moment hypothesis is also inconsistent with the normalization needed to reproduce constants, since k=0 would force ∫χ2=0; that is an additional correctness contradiction rather than a further circular step. Because the central approximation theorem is made true by the proof's inserted normalization, the derivation is partially circular by construction.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

All load-bearing ingredients are external or assumed: the scaling-function partition of unity, smoothness of h, vanishing moments of χ2, and the convolution-kernel bounds from [27]. The only ad hoc number introduced is the 2^{j/2} prefactor, which is not in the definition and is needed to make W an approximant. No new physical entities are postulated.

free parameters (5)
  • 2^{j/2} prefactor = 2^{j/2}
    Inserted in the proof of Theorem 3 but absent from operator definition (1.4). It is the factor that would make the operator an approximation operator, so it is chosen ad hoc to force convergence.
  • K(p, ϱ) = unspecified
    From Theorem 1 (cited as [27]); used in Theorem 4. Its value is never given, and the claimed bound depends on it.
  • K'(p, ϱ) = unspecified
    Absorbed constant in Theorem 4 after applying Hardy-Littlewood-Sobolev; no form or estimate is provided.
  • G(p, ϱ) = unspecified
    From Theorem 2 (cited as [27]); used in Theorem 5. Value never given.
  • G'(p, ϱ) = unspecified
    Absorbed in Theorem 5; no form or estimate.
assumptions (6)
  • domain assumption κ1 is a bounded Haar scaling function with compact support in [−a,a], nonnegative, and Σϱ κ1(t−ϱ)=1
    Used in Theorem 3 to pass from a local integral to sums over the support and to use partition of unity; stated explicitly.
  • standard math h∈C^N(R) or C^m with bounded derivatives, and Taylor's theorem with integral remainder
    Used in proofs of Theorems 3 and 6.
  • ad hoc to paper The integral of χ2 against the shifted kernel behaves like a normalization: effectively ∫χ2(v−ϱ)dv=1
    The proof of Theorem 3 writes ∫ h(v/2^j)χ2(v−ϱ)dv as if χ2 contributes no weight, and the operator is treated as if it contains the sampling-Kantorovich scale factor. This is never stated and is contradicted by the vanishing-moment assumption in Theorem 6.
  • ad hoc to paper χ2 is a wavelet with m vanishing moments: ∫(u−t)^k χ2(2^j u−ϱ)du=0 for k=0,...,m−1
    Stated in Theorem 6; for k=0 it makes ∫χ2=0, incompatible with the normalization needed for convergence.
  • standard math Theorem 1 and Theorem 2 of Pathak and Pathak [27] bounding D(t1,t2,t3)
    Used as black boxes in Theorems 4 and 5; no proof is reproduced.
  • standard math Hardy-Littlewood-Sobolev inequality
    Invoked in Theorems 4 and 5 to estimate convolution with |t−u|^{−ϱ}.

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Pith. "Pith review of On wavelet-based sampling Kantorovich operators and their study in multi-resolution analysis." pith.science (2026). https://pith.science/paper/25E3CWTP

@misc{pith2026250618912,
  author       = {Pith},
  title        = {Pith review of: On wavelet-based sampling Kantorovich operators and their study in multi-resolution analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/25E3CWTP}},
  note         = {Machine review of arXiv:2506.18912}
}
abstract

In this work, wavelet-based filtering operators are constructed by introducing a basic function $D(t_1, t_2, t_3)$ using a general wavelet transform. The cardinal orthogonal scaling functions (COSF) provide an idea to derive the standard sampling theorem in multiresolution spaces which motivates us to study wavelet approximation analysis. With the help of modulus of continuity, we establish a fundamental theorem of approximation. Moreover, we unfold some other aspects in the form of an upper bound of the estimation taken between the operators and functions with various conditions. In that order, a rate of convergence corresponding to the wavelet-based filtering operators is derived, by which we are able to draw some important interferences regarding the error near the sharp edges and smooth areas of the function. Eventually, some examples are demonstrated and empirically proven to justify the fact about the rate of convergence. Besides that, some derivation of inequalities with justifications through examples and important remarks emphasizes the depth and significance of our work.

Figures

Figures reproduced from arXiv: 2506.18912 by the authors.

Figure 1
Figure 1. Graphical representation of E(t) with respect to the function h(t) = sin(2πt). We summarize from [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗

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