Asymptotically best possible Lebesque-type inequalities for the Fourier sums on sets of generalized Poisson integrals

Anatoly Serdyuk; Tetiana Stepaniuk

REVIEW 2 major objections 3 minor 19 references

Asymptotically best possible Lebesque-type inequalities for the Fourier sums on sets of generalized Poisson integrals

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Fourier-sum errors are asymptotically optimal on generalized Poisson classes, for all L_p.

desk verdict 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). read the letter →

arxiv 1908.09517 v1 pith:U7XSI4G4 submitted 2019-08-26 math.CA

Anatoly Serdyuk , Tetiana Stepaniuk This is my paper · ORCID

classification math.CA MSC 42A1041A17

keywords Lebesgue-typeinequalitiesFouriersumsgeneralizedPoissonintegralsbestapproximationsbytrigonometricpolynomialsasymptoticallypossibleestimateshypergeometricfunctionuniformapproximation

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 proves Lebesgue-type inequalities for Fourier sums on sets of functions defined by generalized Poisson integrals. For the previously open case 0

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

The main problem is the algebra between (16) and (20). Estimate (16) has an error term proportional to (1/(αr)^{1+1/p}) n^{-r} , 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.

The 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.

Bottom 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.

Referee Report

2 major / 3 minor

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.

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 (2)

[§2, proof of Theorem 1, Eqs. (16) and (20)/(8)] 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.
[§2, proof of Theorem 2, Eqs. (27), (38) and (42)] 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.

minor comments (3)

[Title] The title contains 'Lebesque', which should be 'Lebesgue'.
[§2, Theorem 1 and Theorem 2] 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.
[End of §2] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation; the Lebesgue inequalities reduce to independent kernel-norm asymptotics and the sharpness proof is a genuine extremal construction.

full rationale

The derivation chain is not circular. The upper bound (8) follows from the exact convolution identity (10), the best-approximation substitution (12)-(13), Hölder's inequality (14)-(15), and the tail-kernel norm asymptotics (16) and (27) quoted from the authors' earlier papers [9] and [10]. Those cited results concern the L_{p'} norms of the kernels P^{(n)}_{α,r,β}; they are parameter-free asymptotic theorems whose assumptions do not include the Lebesgue-type inequality being proved, so citing them is legitimate independent support even though the authors overlap. The sharpness part of Theorem 1 is also non-circular: the extremal function Φ is constructed explicitly in (22), its best approximation is shown to be zero via orthogonality, and the equality (23) is an exact computation of the convolution integral. This realizes the bound rather than assuming it. The final asymptotic-sharpness comparison with Theorem 4 of [9] is an external benchmark, not a self-referential reduction, and the paper also proves equality cases directly through (21)-(23), so it does not depend solely on the comparison. No parameter is fitted to the target constant, and no equation is defined in terms of the claimed result. The algebraic discrepancy between (16) and (20) raised in the skeptical reading, if real, would be a correctness defect in the substitution step, not a circularity, because the theorem would still not be equivalent to its input by construction. Self-citations are numerous and load-bearing, but they point to published, independent theorems, so under the stated rules they do not raise the circularity score.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on standard functional-analytic inequalities plus the kernel-norm asymptotics from the authors' prior works. No free parameters are fitted to data and no new entities are postulated.

assumptions (3)

standard math Convolution norm inequality (14): ||∫ K(t-u)φ(u)du||_C ≤ ||K||_{p'} ||φ||_p for K∈L_{p'}, φ∈L_p.
Invoked at line (14) to pass from the integral representation (10) to the upper bound (15). It is a standard Hölder-type inequality from Korneichuk's book [19, p.43].
standard math Best approximation characterization (Proposition 1.4.12 of [19]): if ∫ t_{n-1} |Φ|^{p-1} sign Φ = 0 for all t_{n-1}∈τ_{2n-1}, then 0 is a best approximation of Φ in L_p.
Used to show t*_{n-1}≡0 for the extremal functions (22) and Φ_ε, which is essential for the sharpness equalities (9) and (25).
domain assumption Tail-kernel asymptotic estimates (16), (27), (35)-(39) from [9] and [10]: two-term asymptotic expansion for ||P^{(n)}_{α,r,β}||_{p'} (1<p<∞) and ||P^{(n)}_{α,r,β}||_∞ (p=1) with error bounded by (14π)^2 for n≥n0(α,r,p).
Directly quoted from the authors' earlier papers; not reproved here. This is the numeric core that produces the leading constant and error bounds in Theorems 1 and 2.

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Cite this review

Pith. "Pith review of Asymptotically best possible Lebesque-type inequalities for the Fourier sums on sets of generalized Poisson integrals." pith.science (2026). https://pith.science/paper/U7XSI4G4

@misc{pith2026190809517,
  author       = {Pith},
  title        = {Pith review of: Asymptotically best possible Lebesque-type inequalities for the Fourier sums on sets of generalized Poisson integrals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U7XSI4G4}},
  note         = {Machine review of arXiv:1908.09517}
}

abstract

In this paper we establish Lebesgue-type inequalities for $2\pi$-periodic functions $f$, which are defined by generalized Poisson integrals of the functions $\varphi$ from $L_{p}$, $1\leq p< \infty$. In these inequalities uniform norms of deviations of Fourier sums $\| f-S_{n-1} \|_{C}$ are expressed via best approximations $E_{n}(\varphi)_{L_{p}}$ of functions $\varphi$ by trigonometric polynomials in the metric of space $L_{p}$. We show that obtained estimates are asymptotically best possible.

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Works this paper leans on

19 extracted references · 19 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [9]

Serdyuk, T.A

A.S. Serdyuk, T.A. Stepanyuk, Approximations by Fourie r sums of classes of generalized Poisson integrals in metrics of spaces Ls, Ukr. Mat. J. 69:5 (2017) 811–822

work page 2017

[2] [8]

Uniform approximatio ns by Fourier sums on the classes of convolutions with generalized Poisso n kernels,

A. S. Serdyuk and T. A. Stepanyuk, “Uniform approximatio ns by Fourier sums on the classes of convolutions with generalized Poisso n kernels,” Dop. Nats. Akad. Nauk. Ukr., No. 11, (2016) 10–16

work page 2016

[3] [10]

Serdyuk, T.A

A.S. Serdyuk, T.A. Stepanyuk, Uniform approximations by Fourier sums on classes of generalized Poisson integrals, Analysis Mathem atica, 45:1 (2018) 201–236

work page 2018

[4] [1]

Stepanets, Methods of Approximation Theory

A.I. Stepanets, Methods of Approximation Theory. VSP: L eiden, Boston 2005

work page 2005

[5] [2]

Stepanets, Deviation of Fourier sums on classes of i nﬁnitely diﬀerentiable functions, Ukr

A.I. Stepanets, Deviation of Fourier sums on classes of i nﬁnitely diﬀerentiable functions, Ukr. Mat. J. 36:6 (1984) 567–573

work page 1984

[6] [3]

Nikol’skii, Approximation of functions in the mean by trigonometrical polynomials, (in Russian) Izv

S.M. Nikol’skii, Approximation of functions in the mean by trigonometrical polynomials, (in Russian) Izv. Akad. Nauk SSSR, Ser. Mat. 10 (1946) 207– 256

work page 1946

[7] [4]

Stechkin, An estimate of the remainder term of Fouri er series for diﬀer- entiable functions, (in Russian) Tr

S.B. Stechkin, An estimate of the remainder term of Fouri er series for diﬀer- entiable functions, (in Russian) Tr. Mat. Inst. Steklova 14 5 (1980) 126–151

work page 1980

[8] [5]

Telyakovskii, Approximation of functions of high s moothness by Fourier sums, Ukr

S.A. Telyakovskii, Approximation of functions of high s moothness by Fourier sums, Ukr. Math. J. 41:4 (1989) 444–451

work page 1989