{"id":"5c9492f8-d1e0-4296-b255-61b95039b142","arxiv_id":"1908.09517","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For generalized Poisson integrals with kernel parameter r in (0,1), the uniform deviation of Fourier sums is bounded above by a sharp constant times the L_p best approximation error of the generalized derivative, for all 1≤p<∞.","lead":"An approximation theory paper proves sharp upper and lower bounds for how far the n-th Fourier sum of a periodic function can deviate from the function, measured uniformly, when the function is a generalized Poisson integral of an L_p function. The result covers the previously open case 0<r<1 and 1≤p<∞, completing a program on Lebesgue-type inequalities.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1's error term is inconsistent with the cited kernel estimate (16); for alpha*r<1 the stated inequality is not supported.","rationale":"The reader identified the reliance on the cited kernel-norm estimates (16) and (27) as the main risk. My pass agrees that those estimates are load-bearing, but finds a more specific and actionable problem inside the proof: the algebra from (16) to (20) changes the exponent of alpha*r in an error term without justification. This is not merely a concern about external correctness; it is an inconsistency within the manuscript's own equations. If the negative exponent in (16) is correct, then the theorem's printed error term is far too small when alpha*r < 1, so the stated inequality (8) is not established. If the positive exponent in (8) is intended, then (16) as quoted is misprinted and the cited estimate needs re-verification. Either way, the manuscript needs a correction before the main theorem can be accepted as stated. The asymptotic best-possible claim is more robust, because the disputed term is o(1) in both readings, so I would not reject the paper outright. A conditional acceptance requiring the authors to fix the exponent and confirm the derivation against a numerical or analytic check of the kernel norm seems proportionate. The p=1 statement (24) uses the negative power (alpha*r)^(-2), which is consistent with the concern being specific to the p>1 formula.","tokens_in":11752,"tokens_out":28802,"duration_ms":281697,"concrete_test":"Compute the tail-kernel norm numerically for one small-alpha*r case: p=2, r=1/2, alpha*r=10^(-6), and n about 10^(17) so that the n0 condition is saturated. Replace the infinite tail by a high-accuracy partial sum over k up to n + 5*n^(1-r), using Euler-Maclaurin or direct summation, and evaluate e^(alpha*n^r) * n^(-(1-r)/p) * (1/pi) * ||P^(n)||_{p'}. Compare the residual above the leading constant with the two candidate error terms (alpha*r)^(-3/2) * n^(-1/2) and (alpha*r)^(3/2) * n^(-1/2). If the residual scales with the negative-power term, then equation (8) as printed is incorrect and needs a corrected exponent.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Equations (16) and (20) are mutually inconsistent in a way that the proof cannot bridge. The kernel-norm estimate quoted from [9] contains the error term 1/((alpha*r)^(1+1/p)) * n^(-r) in equation (16). But equation (20), and consequently the theorem's inequality (8), contains (alpha*r)^(1+1/p) * n^(-r). Equations (17)-(19) only replace the finite integral by the full integral and bound it by p^(1/p'); no algebraic step reverses the exponent of alpha*r. For alpha*r < 1, the two expressions differ by a factor (alpha*r)^(-2-2/p), which can be arbitrarily large. The n0 condition in (7) does not impose a lower bound on alpha*r, so this regime is not excluded. If (16) is the correct estimate from [9], then the quantitative inequality (8) and the matching equality (9) are not established for small alpha*r: the printed n-dependent error term is too small by an arbitrarily large factor. The asymptotic best-possible statement may survive, since both candidate error terms tend to zero as n grows, but the theorem as printed is not proven. This concern is internal to the proof's substitution step, independent of whether the cited estimates are correct.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper establishes Lebesgue-type inequalities for the uniform deviations of Fourier sums on classes of generalized Poisson integrals C^{α,r}_β L_p for 0<r<1 and 1≤p<∞. Theorem 1 (1<p<∞) and Theorem 2 (p=1) bound ||f - S_{n-1}(f)||_C by a leading term e^{-α n^r} n^{(1-r)/p} times an explicit constant plus n-dependent error terms, multiplied by E_n(f^{α,r}_β)_{L_p}. The authors also construct, for each f, a function F with the same best approximation for which the inequality becomes an equality, and they argue that the estimates are asymptotically best possible by comparison with earlier lower bounds. The proofs use convolution norm estimates for the tail kernels P^{(n)}_{α,r,β} quoted from the authors' previous papers [8]–[10], together with an extremal function for p>1 and a two-level step-function construction for p=1.","tokens_in":11948,"tokens_out":13777,"duration_ms":116164,"significance":"If the stated inequalities are correct, the paper fills a previously open case (0<r<1, 1≤p<∞) for Lebesgue-type inequalities on generalized Poisson integrals. The proofs are constructive and give explicit constants, including the bounded error term |γ|≤(14π)^2; the sharpness constructions are explicit and the comparison with [9] and [10] gives a clear asymptotic optimality argument. The main caveats are that the load-bearing tail estimates are quoted without proof and, more importantly, the internal algebra in the proof of Theorem 1 appears inconsistent (see major comments).","major_comments":[{"comment":"Equation (16) quotes from [9] the tail-kernel estimate with error term containing (αr)^{-(1+1/p)} n^{-r} (after absorbing the integral via (19) into p^{1/p'}), whereas equations (20) and (8) contain (αr)^{1+1/p} n^{-r}. The steps (17)–(19) only replace the finite integral by the infinite one and bound it; no algebraic step reverses the sign of the exponent of αr. Since the n0 condition (7) does not restrict αr away from zero, for αr<1 the printed error term in (8) is smaller than the quoted estimate (16) by an arbitrarily large factor (αr)^{-2-2/p}. Consequently, the quantitative inequality (8) and the matching equality (9) are not proven as stated; the authors must either correct (16) or revise (8)/(20) so that the error term is consistent.","section":"§2, proof of Theorem 1, Eqs. (16) and (20)/(8)"},{"comment":"The upper-bound estimate (27) has the error term 1/((αr)^2 n^r) + 1/n^{1-r}, but the lower-bound estimate (38) is derived with the error term 1/(αr n^{1-r}) + 1/n^{1-r} (and (42) uses 1/(αr n^r) + αr/n^{1-r}). These are not algebraically equivalent for general α and r, and the proof does not explain how the constructed function Φ_ε attains a value consistent with the constant in (27). This undermines the claimed equality (25) for the p=1 case; the authors should reconcile the error terms in the upper and lower estimates.","section":"§2, proof of Theorem 2, Eqs. (27), (38) and (42)"}],"minor_comments":[{"comment":"The title contains 'Lebesque', which should be 'Lebesgue'.","section":"Title"},{"comment":"The quantity γ_{n,p} and γ_{n,1} are used in the theorem statements before their bounds are specified; moving the definition of these quantities before the theorems would improve readability.","section":"§2, Theorem 1 and Theorem 2"},{"comment":"The paper states after the proofs that inequalities (8) and (24) were announced in [15]. It would be clearer to say explicitly at the start which results are new proofs rather than new statements.","section":"End of §2"}],"recommendation":"major_revision","confidential_remarks":"The paper's proofs depend on estimates quoted from the authors' earlier papers [9] and [10], which the referee could not independently verify. If those quoted estimates are correct, the algebraic inconsistencies identified in the major comments can be fixed by correcting either the quoted display (16) or the theorem statements. The paper's contribution is incremental but appears genuinely new for the case r∈(0,1), 1≤p<∞, and the explicit constructive sharpness argument is a strength. The referee sees no indication of circularity beyond the heavy reliance on prior work of the same authors."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper fills an honest gap: Lebesgue-type inequalities for generalized Poisson integrals on C^{α,r}_β L_p, r∈(0,1), 1≤p<∞, with asymptotically sharp constants. The earlier literature only had r∈(0,1), p=∞ and r≥1, all finite p. The authors construct the extremal functions explicitly, including a separate adaptive construction for p=1, and they correctly compare their upper bounds to the known asymptotics from their earlier work. The convolution-norm argument is standard and the citation pattern is fine — the load-bearing kernel estimates come from their published papers, which is acceptable since those are independent theorems.\n\nThe main problem is the algebra between (16) and (20). Estimate (16) has an error term proportional to (1/(αr)^{1+1/p}) n^{-r}\n, but the derived expression (20), and therefore Theorem 1's inequality (8), contains (αr)^{1+1/p} n^{-r}. For αr<1 these differ by a factor of (αr)^{-2-2/p}, which can be arbitrarily large. Equations (17)–(19) only replace a finite integral by the full integral and bound it; nothing reverses the power of αr. So as printed, the quantitative inequality (8) is not supported by the cited estimate. This is not a minor typo in a peripheral line — it is the error term of the main theorem. The asymptotic best-possible claim might survive, because the leading term still dominates as n→∞, but the finite-n statement is unproven. The same concern does not apply to Theorem 2, where the p=1 estimate (27) is quoted directly and the sharpness argument uses a separate estimate (38) with consistent powers.\n\nThe p=1 sharpness argument is a bit dense but works: the sign-oscillating function Φ_ε gives the right convolution value up to a small ε-loss, and the ε→0 limit recovers the norm. The comparison for asymptotic sharpness using their own Theorem 4 from [9] is legitimate.\n\nBottom line: the underlying project is sound, the result is likely correct, and the defect is repairable, but the manuscript as submitted would need the author to fix the error-term power and re-verify the constants. A serious referee should see it; it should not be desk-rejected. But my own verdict is that the proof of Theorem 1 has a genuine gap, and the paper needs major revision before acceptance.","headline":"Completes a real parameter range in a niche program, but Theorem 1 has a proof error: the error term in (8)/(20) has the α,r power inverted relative to the cited estimate (16).","tokens_in":12521,"tokens_out":4975,"would_cite":false,"duration_ms":45029,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42A10","41A17"],"pacs":[],"model":"deepseek-v4-flash","headline":"Fourier-sum errors are asymptotically optimal on generalized Poisson classes, for all L_p.","keywords":["Lebesgue-type inequalities","Fourier sums","generalized Poisson integrals","best approximations by trigonometric polynomials","asymptotically best possible estimates","hypergeometric function","uniform approximation"],"falsifier":"Numerically compute (1/π)||$P^{{(n)}}$_{α,r,β}||_{p'} for a specific parameter set such as α=1, r=1/2, β=0, p=2 (so p'=2) over a range of n, and compare with the asymptotic expansion (16): if the difference divided by the remainder terms ever exceeds (14π)^2 for n≥n0, the central estimate fails.","tokens_in":11509,"feed_emoji":"","tokens_out":4642,"duration_ms":43001,"temperature":0.7,"pith_summary":"This paper proves Lebesgue-type inequalities for Fourier sums on sets of functions defined by generalized Poisson integrals. For the previously open case 0<r<1 and 1≤p<∞, it shows that the uniform deviation of a Fourier sum is controlled by the best L_p approximation of the function's generalized derivative, multiplied by an explicit leading constant and a controlled remainder. It also proves these estimates are asymptotically best possible: for every function there is another function with the same best approximation for which the inequality becomes an equality up to the stated remainder. This settles the remaining parameter range and gives computable constants for quantitative error bounds.","feed_headline":"Fourier-sum errors are asymptotically optimal on Poisson classes","feed_subtitle":"New bounds control uniform deviations via L_p best approximations with best-possible constants.","key_machinery":"The argument uses the convolution identity ρ_n(f;x)=(1/π)∫($f^{{α,r}}$_β(t)-t_{n-1}(t))$P^{{(n)}}$_{α,r,β}(x-t)dt, where $P^{{(n)}}$_{α,r,β}(t)=Σ_{k=n}^∞ $e^{{-αk^r}}$cos(kt-βπ/2) is the tail of the generalized Poisson kernel and is orthogonal to trigonometric polynomials of degree <n. A Hölder convolution inequality reduces the uniform error to (1/π)||$P^{{(n)}}$_{α,r,β}||_{p'}E_n($f^{{α,r}}$_β)_{L_p}. The load-bearing estimate is the two-term asymptotic expansion (16) for this tail-kernel norm, taken from the authors' earlier papers, with leading constant expressed through the Gauss hypergeometric function F(1/2,(3-p')/2;3/2;1). Sharpness is shown by constructing an extremal function Φ that attains the norm and has zero best-approximation polynomial: Φ=||$P^{{(n)}}$_{α,r,-β}||_{p'}^{1-p'}|$P^{{(n)}}$_{α,r,-β}|^{p'-1}sign($P^{{(n)}}$_{α,r,-β}) for p>1, and a step-function concentration around a maximum point for p=1.","core_discovery":"The central claim is Theorem 1 (for 1<p<∞) and Theorem 2 (for p=1). For n≥n0(α,r,p), every f in $C^{{α,r}}$_β L_p satisfies ||f-S_{n-1}(f)||_C ≤ $e^{{-αn^r}}$ $n^{{(1-r)/p}}$ [ ||cos t||_{p'} / ($π^{{1+1/p'}}$(αr)^{1/p}) $F^{{1/p'}}$(1/2,(3-p')/2;3/2;1) + γ_{n,p}((...)) ] E_n($f^{{α,r}}$_β)_{L_p}, with |γ_{n,p}|≤(14π)^2, and the same expression with equality holds for some function F with the same E_n. The leading constant is asymptotically best possible on the classes $C^{{α,r}}$_{β,p}, because the same leading term matches known lower bounds from earlier work.","pith_inferences":["The same tail-kernel norm technique could likely be adapted to other kernels with monotonically decaying coefficients, producing analogous sharp Lebesgue inequalities for fractional or multi-parameter Poisson-type integral classes.","The extremal-function construction might transfer to prove sharpness for other linear approximation methods, such as de la Vallée Poussin or Lagrange interpolation sums, on the same generalized Poisson classes.","Tracking constants more carefully in the cited norm estimates could shorten the threshold n0, giving non-asymptotic inequalities valid for smaller values of n.","The sharp constants in these uniform estimates may lead to precise n-width or entropy-number asymptotics for the compact classes C^{α,r}_{β,p} in the uniform metric."],"forward_implications":["The inequalities give explicit, asymptotically sharp constants for uniform Fourier-sum approximation of generalized Poisson integrals in terms of L_p best approximation, covering all 1≤p<∞ and 0<r<1.","On the unit-ball classes C^{α,r}_{β,p}, taking suprema yields upper bounds for E_n(C^{α,r}_{β,p})_C that match previously known asymptotics, proving optimality in the power scale.","The p=1 case yields a sharp bound with leading term (1/(παr))e^{-αn^r} n^{1-r}E_n(f^{α,r}_β)_{L_1}.","The explicit hypergeometric prefactor makes the bounds directly usable for numerical error estimation in applications that rely on Fourier sums for these smooth periodic functions."],"supporting_citations":[{"why":"Supplies the key asymptotic expansion (16) for the L_{p'} norm of the tail kernel P^{(n)}_{α,r,β}, which is the central estimate behind Theorem 1.","marker":"[9]"},{"why":"Provides formula (17) and the estimates (35)-(37) used in Theorem 2 for the L_∞ norm of the tail kernel and the derivative-ratio bound.","marker":"[10]"},{"why":"Supplies the convolution inequality (14) and the best-approximation characterization (Proposition 1.4.12) used to show the extremal function has zero best-approximation polynomial.","marker":"[19]"}],"fun_headline_variants":["Optimal Lebesgue bounds for Fourier sums on Poisson classes","Fourier sums hit best-possible error on Poisson classes","Lebesgue inequalities sharpened for Poisson classes","Fourier approximation: exact leading constant on Poisson sets","Best-possible Fourier errors for Poisson-integral functions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on the previously established estimates (16) and (27) for the L_{p'} and L_∞ norms of the tail kernels, with their stated constants and threshold n0; if those are wrong, the sharp inequalities do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Optimal Lebesgue bounds for Fourier sums on Poisson classes","Fourier sums hit best-possible error on Poisson classes","Lebesgue inequalities sharpened for Poisson classes","Fourier approximation: exact leading constant on Poisson sets","Best-possible Fourier errors for Poisson-integral functions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000253,"raw_usage":{"total_tokens":1507,"prompt_tokens":833,"completion_tokens":674,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":449,"completion_tokens_details":{"reasoning_tokens":594}},"tokens_in":449,"tokens_out":674,"duration_ms":5818,"temperature":1.0,"reasoning_tokens":594,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:09:37.094414+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically compute (1/π)||$P^{{(n)}}$_{α,r,β}||_{p'} for a specific parameter set such as α=1, r=1/2, β=0, p=2 (so p'=2) over a range of n, and compare with the asymptotic expansion (16): if the difference divided by the remainder terms ever exceeds (14π)^2 for n≥n0, the central estimate fails.","supporting_citations":[{"cited_title":"Serdyuk, T.A","cited_arxiv_id":null,"evidence_quote":"Supplies the key asymptotic expansion (16) for the L_{p'} norm of the tail kernel P^{(n)}_{α,r,β}, which is the central estimate behind Theorem 1."},{"cited_title":"Serdyuk, T.A","cited_arxiv_id":null,"evidence_quote":"Provides formula (17) and the estimates (35)-(37) used in Theorem 2 for the L_∞ norm of the tail kernel and the derivative-ratio bound."},{"cited_title":"Korneichuk, Exact Constants in Approximation The ory, Vol","cited_arxiv_id":null,"evidence_quote":"Supplies the convolution inequality (14) and the best-approximation characterization (Proposition 1.4.12) used to show the extremal function has zero best-approximation polynomial."}],"review_version":1}