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On Hermite type sampling Kantorovich operators in the settings of mixed norm Spaces

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Pith's one-line read A Hermite-type sampling Kantorovich operator built from local averages and derivative samples converges uniformly to differentiable signals on $\mathbb{R}\times\mathbb{R}^d$, at rate $w^{-n}$ or better, with a Voronovskaja-type asymptotic…

desk verdict Competent extension of known sampling-operator templates to a new Hermite-Kantorovich combination, but the mixed-norm promise in the title and abstract is unsupported and the simultaneous approximation theorems need a hypothesis fix. read the letter →

arxiv 2506.02468 v1 pith:CV75JU7A submitted 2025-06-03 math.FA

Puja Sonawane , A. Sathish Kumar This is my paper
classification math.FA MSC 41A3594A2041A2526A15
keywords SamplingoperatorsKantorovichHermite-typeMixednormspacesDirectapproximationtheoremsVoronovskaja-typeasymptoticformulaModulusofcontinuitySimultaneous
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

Sampling and reconstruction usually use exact point values of a signal. This paper studies a variant that incorporates both local averages, in the Kantorovich style, and derivative samples up to order $n$, in the Hermite style, for functions of one time variable and $d$ spatial variables, and it analyzes the approximation in the framework of mixed-norm spaces. The central results are uniform convergence to $f$ as the sampling rate $w$ grows, rates of order $w^{-n}$ (or $w^{-n-1}$ with one extra derivative), a Voronovskaja-type asymptotic formula, and simultaneous approximation, in which derivatives of the operator converge to the corresponding derivatives of $f$. A sympathetic reader would care because the results give a template for reconstructing differentiable signals and their derivatives from local average and derivative data, with quantifiable error.

What carries the argument

The central object is the operator $K_{n,w}^{\varphi,\psi} f(x,y) = \sum_{k\in\mathbb{Z}}\sum_{m\in\mathbb{Z}^d}\varphi(wx-k)\prod_i\psi_i(wy_i-m_i)\,w^{d+1}\int_{I_k^w}\int_{I_m^w}\sum_{l+|j|\le n}\frac{1}{l!j!}\frac{\partial^{l+|j|}f(u,v)}{\partial u^l\partial v^j}(x-u)^l(y-v)^j\,du\,dv$. The mechanism is Taylor's formula with derivative terms up to order $n$ expanded about the sample nodes $(k/w,m/w)$, followed by averaging over the cubes $[k/w,(k+1)/w]\times\prod_i[m_i/w,(m_i+1)/w]$; the kernels $\varphi,\psi_i$ are required to satisfy a partition of unity and finite absolute moments. The proof machinery consists of the uniform boundedness of the discrete moments $M_\alpha$, the modulus of continuity for bounding the $n$th-order Taylor remainder, and a differentiation identity that writes $\partial^{p+q}K f$ as a weighted sum of lower-order operators applied to derivatives of $f$. That identity is what turns one convergence theorem into simultaneous approximation.

What would settle it

Take $n=2$, $d=1$, a compactly supported cardinal B-spline kernel, and $f(x,y)=x^2$, which satisfies the paper's definition of $C^2$ (only $\partial^2 f$ is required to be bounded and uniformly continuous) but has unbounded $\partial f/\partial x$. Computing $\|K_{2,w}^{\varphi,\psi} f - f\|_\infty$ for $w=10,100,1000$ would show whether the claimed $o(w^{-2})$ uniform convergence holds under the stated hypothesis or only under the stronger $C^2_b$ hypothesis used in the proofs.

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Extended reading notes

Core claim

The paper's central claim is that for $f \in C^n_b(\mathbb{R}\times\mathbb{R}^d)$, the Hermite-type sampling Kantorovich operator satisfies $\lim_{w\to\infty}\|K_{n,w}^{\varphi,\psi} f - f\|_\infty = 0$, and under stronger smoothness the error is $O(w^{-n})$ or $O(w^{-n-1})$ with an explicit asymptotic: $w^{n+1}(K_{n,w}^{\varphi,\psi} f(x,y) - f(x,y))$ converges to a finite combination of derivatives $\partial^{l+j}f(x,y)$ multiplied by discrete kernel moments $m_c(\varphi)/(l-c+1)$ and $m_{d_i}(\psi_i)/(j_i-d_i+1)$. The same statements hold for derivatives: $\partial^{p+q}K f \to \partial^{p+q} f$ uniformly for $p+|q|\le n$, with matching rates and a Voronovskaja-type expansion. Thus the operator simultaneously reconstructs a differentiable signal and its derivatives from local averages of the signal's derivatives.

Load-bearing premise

The proofs repeatedly bound every mixed derivative $\partial^{l+j}f$ in the supremum norm, so the results depend on all derivatives of $f$ up to order $n$ being bounded and uniformly continuous; the stated hypotheses, which only guarantee this for the top-order derivatives, are weaker than what the arguments use.

Editorial extensions

If this is right

  • For differentiable signals with bounded derivatives, increasing the Hermite order $n$ improves the guaranteed approximation rate from $w^{-1}$ to $w^{-n}$, and one additional derivative buys $w^{-n-1}$.
  • The Voronovskaja-type formula identifies the leading error constant in terms of kernel moments, so kernels can be compared or tuned by their moments.
  • Simultaneous approximation means the same operator provides estimates of $f$ and its derivatives up to order $n$ from the same sample data, useful for numerical differentiation.
  • The numerical implementation with cardinal B-splines shows the predicted improvement: for $w=7$ the reported uniform errors drop from $0.2323$ ($n=0$) to $0.0325$ ($n=1$) to $0.0055$ ($n=2$).

Reading between the lines

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

  • The paper's theorems are uniform-norm statements despite the mixed-norm framing; a genuine $L^{p,q}$ version of the rate and Voronovskaja results is a natural next step that the methods here do not yet provide.
  • The differentiation identity suggests an implementation strategy: one pass of the operator can output $f$ and all its derivatives up to order $n$, which could reduce cost in applications needing gradient or Hessian estimates.
  • The Voronovskaja constant is a function of kernel moments, so it could be used as an objective for kernel design, e.g., choosing the B-spline degree to minimize the leading error for a given smoothness class.
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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

3 major / 5 minor

Summary. This paper proposes a multivariate Hermite-type sampling Kantorovich operator K^{φ,ψ}_{n,w} on R×R^d, obtained by replacing point samples in Corso's Hermite sampling operator with local averages of Taylor polynomials up to order n. The authors prove uniform-norm results: convergence of K f to f for f in C^n_b, rates o(w^{-n}) and O(w^{-n-1}) via moduli of continuity, a Voronovskaja-type asymptotic expansion, and simultaneous approximation of derivatives with corresponding rates and asymptotics. The final section gives numerical experiments with cardinal B-splines. The uniform-norm arguments are standard moment/Taylor estimates.

Significance. If the mixed-norm framing is removed, the paper contributes a natural multivariate Kantorovich version of Corso's Hermite sampling operators with explicit uniform rates and a Voronovskaja formula, including simultaneous approximation. The estimates are given with explicit constants and moment conditions, and the B-spline implementation with numerical tables is a useful check. However, the announced mixed-norm setting is entirely absent from the results, and several theorems are stated under hypotheses weaker than those used in their proofs; these issues must be resolved before the paper's claims are supportable.

major comments (3)
  1. [§1.1, Definition 1, Theorems 2, 4, 5] The statements and proofs are not on the same hypotheses. In §1.1, C^n(R^d) is defined to require ∂^α f ∈ C(R^d) only for |α|=n, so lower-order derivatives need not be bounded. However, Definition 1 involves ∂^{l+j}f for every l+|j|≤n, and the well-definedness estimate preceding Definition 1 bounds ∥∂^{l+j}f∥∞ for all such indices. Theorems 2(i,ii), 4, and 5 state f∈C^n(R×R^d) but their proofs use boundedness and uniform continuity of derivatives of every order up to n. A concrete witness is d=0, n=2, f(x)=x^2: f is in C^2 by the paper's definition, but f' is unbounded, so the quantities ∥∂ K f − f'∥∞ appearing in Theorem 4 are not meaningful and even the definition of K^{φ,ψ}_{2,w}f is not justified by the given well-definedness estimate. The statements and proofs must be harmonized by using C^n_b(R×R^d) (all derivatives of order ≤ n in C_b) throughout.
  2. [Title, Abstract, §1.1, Theorems 1–6] The paper's announced contribution is approximation in mixed norm spaces, but no result in L^{p,q} or ℓ^{p,q} is proved. After defining L^{p,q}(R×R^d) and ℓ^{p,q}(Z×Z^d) in §1.1, the paper proves only sup-norm statements: Theorems 1–6 assert convergence or estimates in ∥·∥∞ or pointwise limits. No inequality of the form ∥K f − f∥_{L^{p,q}} appears, and the 'modulus of continuity in mixed norm settings' promised in the abstract is never defined (Definition 2 is a sup-norm modulus). This is a load-bearing mismatch between the claimed setting and the actual results. The authors must either prove genuine mixed-norm estimates or revise the title, abstract, and Section 1 to describe uniform-norm approximation only.
  3. [Theorems 3 and 6] The Voronovskaja-type results assume 'constant discrete moments m_r(φ,u), m_r(ψ_i,u), r=0,...,n+1, i=1,...,d, for all u∈R and not null,' but this condition is not part of the kernel definition in §1.1 and is not verified for the B-spline kernels used in Section 4. Since these theorems are central to the claimed asymptotic results, the constant-moment condition should be stated as a standing hypothesis on the admissible kernels, and the examples in Section 4 should explicitly check it. As written, the reader cannot tell which concrete kernels satisfy the Voronovskaja hypotheses.
minor comments (5)
  1. [§5, Conclusions] The sentence 'the direct approximation theorems have been proved, including including the uniform convergence theorem' contains a duplicated 'including'.
  2. [§3, Theorem 4 proof] The phrase 'the operator reduces to the usual Knatorovich operator' should read 'Kantorovich operator'.
  3. [§1.1] The notation C(R^d) for 'uniformly continuous and bounded' conflicts with standard conventions, where C usually denotes merely continuous functions; consider using C^0_b or C_u to avoid confusion, and define C^n_b explicitly.
  4. [Definition 1] The interval I^w_k is written as (k/w, k+1/w); this should be the closed interval [k/w, (k+1)/w], and similarly for the product intervals I^w_m.
  5. [Theorem 3] The phrase 'not null' should be 'non-zero' or 'not identically zero', and the moment condition m_r(χ,u)=constant should be introduced together with the kernel definition rather than only in the theorem statement.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the theorems follow from the kernel axioms, Taylor expansion, and moment assumptions; the C^n versus C^n_b hypothesis gap is an assumption mismatch, not circularity.

full rationale

The derivation chain is self-contained. Definition 1 defines K^{phi,psi}_{n,w} directly from the kernel and derivatives, and the well-definedness estimate before it explicitly requires all partial derivatives up to order n to be bounded. Theorem 1 uses Taylor's formula with remainder of order n together with the kernel order-n moment finiteness to obtain uniform convergence; the target result is not assumed. Theorem 2(i)-(iii) derive o(w^{-n}) and O(w^{-n-1}) estimates from the uniform continuity of the order-n derivatives and from the discrete moment conditions, while Theorem 3 obtains the Voronovskaja limit by expanding the remainder terms and using the constant-moment hypotheses; no prior theorem is invoked to force the limit. The simultaneous approximation results in Section 3 are Leibniz-rule identities (Lemma 1) combined with the already-proved base cases, so they do not reduce to their own conclusions. References to earlier work by Butzer, Corso, Costarelli, Vinti and others are external context and kernel constructions, not load-bearing self-citations, and there is no imported uniqueness theorem or ansatz from the authors' own prior papers. The only notable defect is a statement-proof mismatch: C^n is defined in Section 1.1 to require only that derivatives of exact order n lie in C(R^{d+1}), whereas Theorems 2, 4 and 5 are stated for f in C^n but their proofs use ||partial^{l+j}f||_infinity for every l+|j| <= n, and Theorem 4 even asserts sup-norm convergence of partial^{p+q}K f, which presupposes boundedness of that derivative. The proofs evidently require C^n_b, as Proposition 1 and Theorem 1 explicitly state. This is a correctness or hypothesis-clarification issue, not circularity: strengthening the assumption does not rename an input as a prediction, and the estimates remain derived rather than assumed. Therefore the circularity score is 0.

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

The analysis rests on standard kernel axioms and smoothness assumptions; no data are fitted and no new physical or mathematical entities are postulated.

assumptions (6)
  • domain assumption Kernel of order n: partition of unity sum_k chi(u-k)=1 and uniform convergence of sum_k |u-k|^n |chi(u-k)|, with finite absolute moments M_r.
    This is the core premise of the entire analysis; it ensures the operator is well-defined and that moment-tail arguments converge (Definition 1, Remark 1).
  • domain assumption Function smoothness: f in C^n_b(R times R^d), meaning all partial derivatives up to order n exist, are continuous, and are bounded.
    The well-definedness estimate and all proofs use ||partial^{l+j} f||_infinity for l+|j| <= n. The notation C^n in Section 1.1 is weaker, creating a gap.
  • standard math Multivariate Taylor formula with Lagrange remainder at possibly distinct intermediate points in the box [u,x] times [v,y].
    Used in Theorems 1, 2, 3, and 6 to expand f(x,y) around (u,v). The paper does not justify the multivariate remainder, though it is a standard iteration of the univariate Taylor theorem.
  • ad hoc to paper Constant discrete moments m_r(chi,u) independent of u and not identically null, for r=0,...,n+1.
    Assumed in Theorems 3 and 6 for the Voronovskaja formula. The paper gives no examples satisfying this condition, and it fails for the B_2 kernel used in Section 4 beyond first order.
  • domain assumption Uniform convergence of derivative series sum_k |u-k|^n |chi^{(l)}(u-k)| for l=1,...,n.
    Required in Proposition 1, Lemma 1, and Theorems 4, 5, 6 to justify term-by-term differentiation of the operator series.
  • domain assumption Finite higher-order absolute moments M_{n+1}(phi), M_{n+1}(psi_i) < infinity.
    Used in Theorem 2(ii)(iii) and Theorem 5 to obtain rates involving the (n+1)-th moments.

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Pith. "Pith review of On Hermite type sampling Kantorovich operators in the settings of mixed norm Spaces." pith.science (2026). https://pith.science/paper/CV75JU7A

@misc{pith2026250602468,
  author       = {Pith},
  title        = {Pith review of: On Hermite type sampling Kantorovich operators in the settings of mixed norm Spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CV75JU7A}},
  note         = {Machine review of arXiv:2506.02468}
}
abstract

In this paper, we analyze the convergence behavior of Hermite-type sampling Kantorovich operators in the context of mixed norm spaces. We prove certain direct approximation theorems, including the uniform convergence theorem, the Voronovskaja-type asymptotic formula, and an estimate of error in the approximation in terms of the modulus of continuity in mixed norm settings. Next, we estimate the rate of convergence of these sampling Kantorovich operators in terms of the modulus of continuity. In addition, we obtain simultaneous approximation results of these sampling Kantorovich operators, including the uniform approximation, the asymptotic formula, and the approximation error in terms of the modulus of continuity in mixed settings. Finally, using cardinal $B$-splines, the implementation of differentiable functions has been shown.

Figures

Figures reproduced from arXiv: 2506.02468 by the authors.

Figure 1
Figure 1. The function f Now consider the following kernels: ϕ(x) = B2(x), ψ(y) = B2(y), x, y ∈ [−2, 2] [PITH_FULL_IMAGE:figures/full_fig_p023_1.png] view at source ↗
Figure 2
Figure 2. The approximation of f by K ϕ,ψ 0,7 f [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗
Figure 3
Figure 3. The approximation of f by K ϕ,ψ 1,7 f [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: The approximation of f by K ϕ,ψ 2,7 f According to Theorem 3, increasing the order leads to a better approximation, which is reflected in the numerical errors. ∥K ϕ,ψ 0,7 f − f∥∞= 0.2323, ∥K ϕ,ψ 1,7 f − f∥∞= 0.0325, ∥K ϕ,ψ 2,7 f − f∥∞= 0.0055 [PITH_FULL_IMAGE:figures/…
Figure 5
Figure 5. Figure 5: illustrates the errors En(w) = ∥Kϕ,ψ n,wf − f∥∞ as functions of w for n = 1, 2, along with the corresponding bounds Tn(w) = 2 n−1 wn+1 X l+j=n+1 1 l! j! ∥∂ l+j f∥∞(Ml(ϕ) + M0(ϕ))(Mj (ψ) + M0(ψ)), as stated in part (iii) of Theorem 2. Comparing the two figures, we obser…
Figure 6
Figure 6. Figure 6: The function fxy Plots of the approximation ∂ 2K ϕ,ψ 3,w f ∂x∂y of fxy for w = 3, 7, 12 are shown in figure 7, figure 8, and figure 9, respectively [PITH_FULL_IMAGE:figures/full_fig_p026_6.png]
Figure 7
Figure 7. Figure 7: The approximation of fxy by (K ϕ,ψ 3,3 f)xy [PITH_FULL_IMAGE:figures/full_fig_p026_7.png]
Figure 8
Figure 8. Figure 8: The approximation of fxy by (K ϕ,ψ 3,7 f)xy [PITH_FULL_IMAGE:figures/full_fig_p027_8.png]
Figure 9
Figure 9. Figure 9: The approximation of fxy by (K ϕ,ψ 3,12f)xy As w increases, approximation errors will reduce. This can be checked in [PITH_FULL_IMAGE:figures/full_fig_p027_9.png]

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