{"id":"304afbbd-67a5-4024-8c28-613307cc4e5b","arxiv_id":"2411.15475","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A nonlinear exponential Kantorovich sampling operator is introduced, with convergence theorems, quantitative error bounds, and a Voronovskaja-type formula in Mellin-Orlicz spaces.","lead":"The paper introduces a nonlinear version of the exponential Kantorovich sampling series and proves that it converges pointwise and uniformly, with explicit rates, plus a Voronovskaja-type asymptotic bound. It extends these results to Mellin-Orlicz spaces, giving quantitative error estimates for a wider class of functions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The printed (L2) moment condition is inconsistent with Lemma 3(ii), on which the uniform tail estimate in Theorem 6 rests; the intended scaling must be corrected before the central convergence claim is verifiable.","rationale":"The reader's weakest-assumption analysis focused on condition (χ4), which is indeed a load-bearing constant-reproduction hypothesis. However, a more immediate obstruction appears in the printed statement of (L2) and Lemma 3. The central convergence proof of Theorem 6 decomposes the error and sends the tail I_{1,2} to zero using Lemma 3(ii). That lemma is only valid under a moment condition on |w ln x - t_k|, not on |ln x - t_k|, for the scaling written in the manuscript. If the printed formulas are taken literally, the uniform convergence result does not follow from the stated assumptions; if they are typographical, the manuscript needs a corrected statement so that a referee can verify the chain. This is a concrete, checkable consistency issue, not a challenge to the underlying method. I therefore recommend a conditional acceptance: the paper should be accepted once the kernel scaling and moment definition are corrected and Lemma 3 is re-verified under those corrected definitions.","tokens_in":51,"tokens_out":35283,"duration_ms":525460,"concrete_test":"Independently re-derive Lemma 3(ii) from the printed (L2) definition. If the derivation requires replacing |ln x - t_k|^β with |w ln x - t_k|^β, or replacing e^{-t_k} x w with e^{-t_k/w} x, then the current statement of (L2) is wrong. As a numerical check, take L(y)=y e^{-y}, t_k=k, w=10^2,10^3,10^4, γ=1, and compare the ratio of the left side of Lemma 3(ii) to M_{β,Π}(L)/(γ^β w^β) under the printed definitions; if the ratio grows with w, the lemma is false as stated.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 2 defines M_{β,Π}(L) = sup_x Σ_k L(e^{-t_k} x w) |ln x - t_k|^β, while Lemma 3(ii) asserts Σ_{|t_k - w ln x| > γw} L(e^{-t_k} x w) ≤ (γ^β w^β)^{-1} M_{β,Π}(L). As printed, these two statements are not compatible. The tail condition controls |t_k - w ln x|, but the moment condition controls |ln x - t_k|; no printed inequality relates these two quantities. With the natural scaling, the kernel argument should be e^{-t_k/w} x and the moment should involve |w ln x - t_k|^β, or an equivalent shift. As it stands, Lemma 3(ii) does not follow from the stated assumptions. Theorem 6 estimates the tail term I_{1,2} by exactly Lemma 3(ii); without this estimate, the uniform part of the central convergence theorem is unsupported. This is not a disagreement about rates but an internal inconsistency in the stated framework. The intended argument is likely salvageable by correcting the scaling, but the printed version of the assumption must be fixed.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the nonlinear exponential Kantorovich sampling series, defined by replacing the sample values in the nonlinear exponential sampling series of Costarelli [22] with Steklov-type averages. The main results are: pointwise and uniform convergence for bounded functions at continuity points and for log-uniformly continuous bounded functions (Theorem 6); quantitative convergence estimates in terms of the log-modulus of continuity (Theorem 7); a Voronovskaya-type asymptotic formula in the form of a limsup estimate under Mellin differentiability (Theorem 9); modular convergence in Mellin-Orlicz spaces via a density argument (Theorems 11 and 13) and a modular inequality (Theorem 12); and quantitative modular estimates with rates (Theorem 14 and Corollary 15). The proofs use standard machinery, including Jensen's inequality, Vitali's convergence theorem, Fubini-Tonelli, and a density argument, and all assumptions on the nonlinear kernel and the auxiliary function L are stated explicitly.","tokens_in":20742,"tokens_out":18067,"duration_ms":141358,"significance":"If the technical issues identified below are corrected, the paper constitutes a substantial and useful generalization of the linear exponential Kantorovich sampling theory to a nonlinear framework, including results in Mellin-Orlicz spaces. The exposition is methodical, the hypotheses are explicit, and there are no fitted parameters: every theorem is derived from stated assumptions on the kernel and the class of functions, and the quantitative rates are concrete enough to be checked for specific kernels. The main weakness is a scaling and notation inconsistency in the statement of condition (L2) and in several changes of variables, which affects the proof of the central convergence theorem; the intended corrections are evident from the proofs themselves, so the results are likely salvageable without changing their scope.","major_comments":[{"comment":"Condition (L2) as printed defines M_{β,Π}(L) = sup_x Σ_k L(e^{-t_k x w}) |ln x - t_k|^β, but Lemma 3(ii) concludes Σ_{|t_k - w ln x| > γw} L(e^{-t_k x w}) ≤ (γ^β w^β)^{-1} M_{β,Π}(L). These two statements are incompatible: the tail condition controls |t_k - w ln x|, while the printed moment controls |ln x - t_k|, and the kernel argument e^{-t_k x w} is not invariant under the shift implicit in the tail. Consequently, Lemma 3(ii) does not follow from (L2) as stated. This is load-bearing because Theorem 6 estimates the uniform tail term I_{1,2} exclusively through Lemma 3(ii), and Theorems 7 and 9 use the same moment in the form |w ln x - t_k|^β. The intended definition is evidently M_{β,Π}(L) = sup_x Σ_k L(e^{-t_k/w} x) |w ln x - t_k|^β, or an equivalent w-normalized form; with this correction Lemma 3(ii) is immediate. The printed version must be corrected, and the kernel argument notation made unambiguous throughout.","section":"Section 2, condition (L2); Lemma 3; Theorem 6"},{"comment":"The proof of Lemma 10 uses the substitution w ln x - t_k = ln y, which is only compatible with a kernel argument of the form x e^{-t_k/w} and a moment involving |w ln x - t_k|. With the printed kernel argument e^{-t_k x w} and moment |ln x - t_k|, the displayed inequality ∫_{|ln x|>M} w L(e^{-t_k x w}) dx/x ≤ ∫_{|ln y|>(M-γ)w} L(y) dy/y is not justified. Since Lemma 10 is used in Theorem 11 for the Vitali tail estimate, this gap must be closed by consistent notation. Similarly, in the proof of Theorem 12 the identity ∫_0^∞ L(e^{-t_k x w}) dx/x = ||L||_{1,μ}/w is asserted; under the intended kernel argument x e^{-t_k/w} the integral equals ||L||_{1,μ}, so the factor 1/w is spurious. The final modular inequality is correct only after cancellation of this factor, so the proof must be rewritten in a way that makes the change of variables and the resulting constants transparent.","section":"Lemma 10 and the proof of Theorem 12"}],"minor_comments":[{"comment":"The notation e^{-t_k x w} is typographically ambiguous and is used inconsistently with the changes of variables in the proofs; the final version should use an unambiguous form such as e^{-t_k/w} x or e^{-t_k} x w and keep it consistent across definitions, lemmas, and theorems.","section":"Throughout"},{"comment":"The definition of M_{β,Π}(L) should be stated with |w ln x - t_k|^β to match Lemma 3(ii) and the estimates in Theorem 7, where the moment M_{1,Π}(L) is used with |w ln x - t_k|.","section":"Section 2, condition (L2)"},{"comment":"In the proof of Theorem 14, the text 'uniformly with respect to x ∈ R^n' should read x ∈ R_+, since the domain throughout the paper is R_+.","section":"Theorem 14, proof"},{"comment":"The bibliographic entry for [22] appears incomplete: the venue is listed as 'Results of Mathematical Analysis and its Applications' without a journal or publisher; please check and complete the reference.","section":"Reference [22]"},{"comment":"There is a typo in the abstract: 'continuous function s' should read 'continuous functions'; the whole paper would benefit from a final proofreading pass for such rendering artifacts.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The central results are plausible and the paper is well within the scope of the journal, but the scaling inconsistency in condition (L2) and in several changes of variables is a genuine barrier to verification. The intended fix is clear from the proofs—replace |ln x - t_k| by |w ln x - t_k| and use the kernel argument x e^{-t_k/w} consistently—so a careful revision should be sufficient. I would not recommend rejection on the basis of the identified issues."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper introduces the nonlinear exponential Kantorovich sampling operator K^χ_w f, where a nonlinear kernel χ acts on Steklov averages of f over exponentially spaced intervals. It proves pointwise and uniform convergence (Theorem 6), quantitative estimates via the log-modulus of continuity (Theorem 7), a limsup-type Voronovskaja formula (Theorem 9), and modular convergence plus rates in Mellin-Orlicz spaces (Theorems 11–14). This is a natural next step after Costarelli's nonlinear exponential sampling and the linear Kantorovich exponential series, and the results are genuinely new, not a repackaging.\n\nThe proofs are competent and use standard machinery: Jensen, Vitali, Fubini-Tonelli, density arguments. Conditions are explicit and no fitted parameters appear. The modular convergence part in particular is a useful extension for signals that are not necessarily log-uniformly continuous.\n\nThe soft spot is real and matches the stress-test note. In Section 2, condition (L2) defines M_{β,Π}(L) = sup_x ∑_k L(e^{-t_k} x w) |ln x − t_k|^β, while Lemma 3(ii) asserts a tail bound over |t_k − w ln x| > γw using that same moment. These two are not compatible: the moment controls |ln x − t_k|, the tail controls |t_k − w ln x|. In the proofs of Theorems 7 and 9, however, the authors consistently use |w ln x − t_k|^β and the kernel argument e^{-t_k} x^w, not e^{-t_k} x w. So the intended definition is clear: the printed (L2) is a typo, almost certainly a lost w in the moment factor. But as printed, the central tail estimate in Theorem 6 is unsupported. This is a must-fix, not a fatal flaw; the framework is salvageable and the rest of the paper reads as correct.\n\nTwo smaller caveats: the Voronovskaja formula is a limsup estimate, which is weaker than a true asymptotic expansion; the authors are upfront about this, so it is a limitation rather than an error. And the plain-text equation rendering is garbled in enough places that a referee will need the actual LaTeX to line-check the constants.\n\nWho should read this: approximation theorists working on nonlinear sampling operators and Mellin analysis. It is a solid incremental contribution, not a breakthrough, but it extends the theory in a useful direction. With the (L2) typo corrected and the LaTeX made available, this deserves a serious referee and likely acceptance after revision.","headline":"A solid, genuine extension of nonlinear exponential sampling to Kantorovich averages; the main issue is a likely typo in the moment condition (L2), which must be fixed before the central convergence claim holds as stated.","tokens_in":21271,"tokens_out":8713,"would_cite":true,"duration_ms":71827,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["41A25","41A35","46E30","47A58","47B38"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that nonlinear exponential Kantorovich sampling series reconstruct bounded functions at their continuity points, uniformly for log-uniformly continuous functions, with quantitative rates and a Voronovskaja-type formula.","keywords":["nonlinear exponential sampling","Kantorovich sampling operators","log-modulus of continuity","Mellin derivative","Voronovskaja formula","Mellin-Orlicz spaces","logarithmic Haar measure","modular convergence"],"falsifier":"Evaluate the operator on the constant function $f\\equiv 2$ with a kernel of the form $\\chi(x,u)=L(x)u^2$, where $L$ satisfies $(L1)$-$(L2)$ and is normalized so that $\\sum_{k\\in\\mathbb{Z}}L(e^{-t_k}xw)\\to 1$. Direct calculation gives $(K_w^\\chi 2)(x)\\to 4$, not $2$, so this kernel fits the Lipschitz structure but fails $(\\chi4)$; observing that failure confirms that the constant-reproduction condition is exactly what stops the operator from converging to a distorted signal.","tokens_in":20321,"feed_emoji":"📈","tokens_out":10977,"duration_ms":88719,"temperature":0.7,"pith_summary":"The paper introduces the nonlinear exponential Kantorovich sampling series, in which the usual point samples of a function are replaced by local averages over short logarithmic intervals and then passed through a nonlinear kernel. Its central claim is that these operators reproduce bounded functions: pointwise at every continuity point, and uniformly when the function is also log-uniformly continuous, provided the kernel satisfies a uniform constant-reproduction condition. Working from that condition and the kernel's moment bounds, the paper obtains quantitative error estimates in terms of the log-modulus of continuity, algebraic rates for locally log-Hölderian functions, a Voronovskaja-type asymptotic formula expressed through the Mellin derivative, and a modular-convergence theory in Mellin-Orlicz spaces with respect to the logarithmic Haar measure. A reader should care because this supplies a nonlinear analogue of a standard linear sampling tool, with concrete rates inherited from the kernel rather than from the target function's smoothness alone.","feed_headline":"Bounded signals recovered by nonlinear sampling at continuity points","feed_subtitle":"Bounded, log-uniformly continuous signals are recovered at explicit rates; the proof extends to Mellin-Orlicz spaces.","key_machinery":"The central object is the operator $K_w^\\chi f(x)=\\sum_{k\\in\\mathbb{Z}}\\chi(e^{-t_k}xw,\\, \\frac{w}{\\Delta_k}\\int_{t_k/w}^{t_{k+1}/w} f(e^u)\\,du)$, a Kantorovich (averaged-sample) version of the nonlinear exponential sampling series. It is carried by the bivariate kernel $\\chi(x,u)$, assumed $(L,\\psi)$-Lipschitz in $u$, and by the moment conditions $(L1)$-$(L3)$ on the amplitude function $L$. The decisive assumption is $(\\chi4)$: the kernel sums $\\sum_{k\\in\\mathbb{Z}}\\chi(e^{-t_k}xw,u)$ must approximate $u$ with error $O(w^{-\\alpha})$ uniformly in $x$, both for small $|u|<1/j$ and for $|u|\\ge 1/j$; in the Orlicz section a stronger one-sided version $(\\chi4^*)$ is used. This constant-reproduction property separates the operator's action on $f$ from its action on the constant $f(x)$, and the proof splits the error into a continuity part controlled by the log-modulus of continuity and a kernel part controlled by $(\\chi4)$.","core_discovery":"The discovery is a convergence theorem that transfers the known linear exponential Kantorovich sampling theory to a nonlinear kernel setting. For any bounded $f$ and any nonlinear kernel $\\chi$ satisfying conditions $(\\chi1)$-$(\\chi4)$, the operator $K_w^\\chi f$ converges to $f$ at every point of continuity, and if $f$ is bounded and log-uniformly continuous then the convergence is uniform on $\\mathbb{R}_+$. The same mechanism yields a quantitative estimate: with a concave $\\psi$, the uniform error is bounded by a combination of $\\psi$ applied to the log-modulus of continuity of $f$ and a $w^{-\\alpha}$ term coming from the kernel's constant-reproduction error. Under local log-Hölderian smoothness this becomes an algebraic rate $w^{-\\min\\{\\nu q,\\alpha\\}}$ or $w^{-\\min\\{\\nu\\beta q,\\beta,\\alpha\\}}$ depending on which absolute moments of the kernel are finite. For functions that are Mellin differentiable at a point, a limsup Voronovskaja formula bounds $w^r|K_w^\\chi f(x)-f(x)|$ by constants involving $|(\\theta f)(x)|^r$, the Mellin derivative. In Mellin-Orlicz spaces the same operator is shown to be modularly convergent for convex $\\phi$ satisfying a compatibility condition $(H)$, with quantitative estimates in terms of the log-modulus of smoothness.","pith_inferences":["The uniform-in-$x$ form of $(\\chi4)$ suggests that the same convergence argument applies to vector-valued or multivariate signals by applying the scalar proof componentwise, as long as the $\\psi$-Lipschitz condition holds in the chosen norm; the paper does not state this extension.","A natural testable strengthening of the Voronovskaja result would be to replace the limsup by a genuine limit under finer assumptions on the kernel's local moments; the present theorem identifies the correct order of magnitude but not a limiting constant.","Remark 5 relaxes boundedness to logarithmic growth $|f(e^x)|\\le a+b|x|$ when $\\psi$ is the identity; pushing this through the modular argument would give convergence for unbounded but slowly growing signals, which the paper does not develop into a theorem.","The stronger condition $(\\chi4^*)$ used in the Orlicz quantitative estimates is not equivalent to $(\\chi4)$; constructing a kernel that satisfies $(\\chi4)$ but fails $(\\chi4^*)$ would determine exactly where the quantitative modular rates in Section 5 break down."],"forward_implications":["For every bounded function, the nonlinear exponential Kantorovich series converges pointwise at each continuity point; for bounded log-uniformly continuous functions the convergence is uniform.","The quantitative estimates provide explicit rates: if $f$ is locally log-Hölderian of order $\\nu$ and $\\psi(u)=O(u^q)$ near zero, the uniform error is $O(w^{-\\min\\{\\nu q,\\alpha\\}})$, with the exponent adjusted to $\\min\\{\\nu\\beta q,\\beta,\\alpha\\}$ when only fractional moments of $L$ exist.","At any point where the Mellin derivative $(\\theta f)(x)=xf'(x)$ exists, the limsup of $w^r|K_w^\\chi f(x)-f(x)|$ is no larger than $(\\Delta^r/2^r)M_{0,\\Pi}(L)|(\\theta f)(x)|^r + M_{r,\\Pi}(L)|(\\theta f)(x)|^r$, for $0