Cesaro Means along Polynomial Subsequences of Fourier Partial Sums at Lebesgue Points

Ushangi Goginava

Cesaro Means along Polynomial Subsequences of Fourier Partial Sums at Lebesgue Points

T0 review · 0 major / 4 minor · reviewed 2026-08-27 · deepseek-v4-flash

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Pith's one-line read For every integrable function on the circle, the (C,α) means of its Fourier partial sums along any sequence that eventually equals an integer-valued polynomial of degree at least two converge to the function at every Lebesgue point.

desk verdict A short, correct-looking proof that settles Zalcwasser's 1936 questions in stronger pointwise form; the only soft spot is a compressed Weyl estimate that survives scrutiny. read the letter →

arxiv 2608.14008 v1 pith:75H5NSHV submitted 2026-08-14 math.CA

Ushangi Goginava This is my paper · ORCID

classification math.CA MSC 42A2442A2011L0740G05

keywords FourierseriessubsequentialpartialsumsCesàromeansLebesguepointspolynomialsubsequencesexponential(Calpha)fractional-order

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

The paper proves that, for every exponent $0<\alpha\le 1$ and every integer-valued polynomial $P$ of degree $r\ge 2$ with positive leading coefficient, the $(C,\alpha)$ means of the Fourier partial sums along the subsequence $a_k=P(k)$ converge to $f(x)$ at every Lebesgue point $x$ of every $f\in L^1(T)$. This answers, in a stronger pointwise form, two questions posed in 1936: whether arithmetic means along higher powers $k^r$ converge for $r>2$, and whether fractional-order Cesàro means behave the same way. The proof represents the means by an integral kernel and shows that the kernel is dominated by an integrable majorant via a uniform polynomial Weyl estimate. If correct, the theorem extends the classical Lebesgue-point summability of full Fourier partial sums to a broad family of polynomial subsequences.

What carries the argument

The central object is the kernel $$K_{N,\$\alpha$}(t)=\frac{1}{$A^{{\alpha}}$_{N-1}}\sum_{k=1}^{N}$A^{{\alpha-1}}$_{N-k}\frac{\sin((a_k+1/2)t)}{2\sin(t/2)},$$ which converts the difference between the $(C,\alpha)$ mean and $f(x)$ into an integral. The argument is carried by Lemma 2, a polynomial Weyl estimate: for a polynomial $Q$ of degree $r$ with leading coefficient $ct$, the exponential sum $\sum_{j=0}^{L-1}e^{iQ(j)}$ is bounded by $C(Lt^{\eta}+L^{1-\eta}+t^{-\eta}L^{1-r\eta})$ with a constant independent of the lower-order coefficients of $Q$. Lemma 3 then turns the existence of a nonincreasing majorant $M_N$ satisfying a bounded-integral condition into convergence at every Lebesgue point. The two lemmas together reduce the theorem to checking two inequalities for $M_N$ built from the Weyl estimate.

What would settle it

A single concrete counterexample would settle the matter: an admissible polynomial $P$, an exponent $\alpha\in(0,1]$, a function $f\in L^1(T)$, and a Lebesgue point $x$ of $f$ for which $\sigma^a_{N,\alpha}f(x)$ does not converge to $f(x)$. Numerically, one can test a non-monomial case, for instance $P(k)=k^2+k$, $\alpha=1/2$, and $f$ the indicator of an interval with $x$ in its interior, and compute the means at large $N$; any persistent discrepancy would refute Theorem 1.

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Extended reading notes

Core claim

The central claim is that the subsequential Cesàro means $\sigma^a_{N,\alpha}f(x)$ tend to $f(x)$ at every Lebesgue point whenever the index sequence satisfies $a_k=P(k)$ for all sufficiently large $k$, with $P$ an integer-valued polynomial of degree $r\ge 2$ and positive leading coefficient. The argument writes $\sigma^a_{N,\alpha}f(x)-f(x)$ as an integral against the kernel $K_{N,\alpha}(t)$, splits the integral into a small interval where the Lebesgue-point condition controls the growth, an intermediate interval controlled by a majorant $M_N$, and a tail handled by the decay of Fourier coefficients and a weighted average argument. The proof's engine is the polynomial Weyl estimate, whose constant is independent of the lower-order coefficients of the shifted polynomial $u\mapsto P(N-u)$; that uniformity is what makes the majorant $M_N$ satisfy the bounded-integral condition.

Load-bearing premise

The load-bearing premise is the polynomial Weyl estimate's uniformity: the bound must hold with one constant even though the lower-order coefficients of the shifted polynomial $P(N-u)$ change with $N$; if that uniformity failed, the kernel could not be dominated by a single integrable majorant and the proof would collapse.

Editorial extensions

If this is right

For $P(k)=k^r$, the $(C,\alpha)$ means along the powers $k^r$ converge at every Lebesgue point for every $r\ge2$ and $0<\alpha\le1$, answering both 1936 questions affirmatively.
At $\alpha=1$, the arithmetic means along any polynomial subsequence of degree at least two converge at every Lebesgue point, strengthening the almost-everywhere convergence known for squares.
Every $f\in L^1(T)$ is covered with no extra regularity assumption; because Lebesgue points have full measure, almost-everywhere convergence is an immediate corollary.
Only the eventual values of the sequence matter, so a finite initial modification of the subsequence does not change the conclusion.
The theorem applies to all integer-valued polynomials with positive leading coefficient, not only monomials, making the result genuinely broader than the original higher-power question.

Reading between the lines

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

An extension not stated in the paper: the same kernel-majorant scheme may tolerate bounded perturbations of the polynomial, such as $a_k=P(k)+O(1)$, because the Weyl estimate's uniformity should absorb lower-order changes; this is a conjecture, not a theorem here.
A possible direction: the majorant lemma is formulated for one-dimensional Fourier series, but its logic is dimension-free, so analogous polynomial-subsequence $(C,\alpha)$ means on higher-dimensional tori may hold; the paper does not address this.
The role of the degree condition $r\ge2$ suggests that genuine lacunarity is not needed: the convergence mechanism relies on oscillation of the exponential sums, not on gaps between indices. Whether the statement holds for sequences like $a_k=k+\log k$ is left open.

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Formalized claims in Lean

Claim #1: The central claim is that the subsequential Cesàro means $\sigma^a_{N,\alpha}f(x)$ tend to $f(x)$ at every Lebesgue point whenever the index sequence satisfies $a_k=P(k)$ for all sufficiently large $k$, with $P$ an integer-valued polynomial of degree $r\ge 2$ and positive leading coefficient. The argument writes $\sigma^a_{N,\alpha}f(x)-f(x)$ as an integral against the kernel $K_{N,\alpha}(t)$, sp

/-- @claim 1 The central claim is that the subsequential Cesàro means $\sigma^a_{N,\alpha}f(x)$ tend to $f(x)$ at every Lebesgue point whenever the index sequence satisfies $a_k=P(k)$ for all sufficiently large $k$, with $P$ an integer-valued polynomial of degree $r\ge 2$ and positive leading coefficient. The argument writes $\sigma^a_{N,\alpha}f(x)-f(x)$ as an integral against the kernel $K_{N,\alpha}(t)$, sp -/
def central_claim : Prop :=

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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

I read Goginava's arXiv:2608.14008. The headline: the paper resolves both Zalcwasser questions from 1936, and it does so at every Lebesgue point instead of almost everywhere. That is a real result, not a repackaging of known tools. The main theorem — (C,α) means of Fourier partial sums along P(k), for integer-valued polynomial P of degree r≥2 with positive leading coefficient, converge at all Lebesgue points for all α∈(0,1] — directly covers the two open cases: higher powers k^r and fractional Cesàro orders.

The proof is short and mostly clean. The kernel representation (2.3) plus the standard majorant-and-Toeplitz machinery in Lemma 3 is a solid framework. The load-bearing part is Lemma 2, the polynomial Weyl estimate. The proof as written is compressed: it cites van der Corput in [5], asserts the reduction to (2.4), and says the constants are uniform. That is the right place to be suspicious. I checked the uniformity claim carefully: the r-th derivative of Q is r! c t, so the van der Corput bound depends only on |c|t, and the lower-order coefficients drop out. The case split tL^r < 1 and ≥1 works, and for r=2, ρ=1/2 gives the borderline exponent. The estimate (2.4) holds as stated with constants uniform in lower-order coefficients. So the soft spot is real in presentation but not in substance.

The later Abel summation in (2.16), the bound (2.17), and the majorant M_N in (2.18) all check out. The asymptotic a_N ≍ N^r supplies exactly the right scaling for condition (2.6). The Toeplitz step for the final interval is standard. I found no circularity and no invented entities; the citation pattern is appropriate, resting on Graham–Kolesnik, Zygmund, and the earlier Belinsky/Izumi–Kawata/Gát work.

Minor: the abstract has a typo — "polynomial of degree with positive leading coefficient" should read "degree r". Nothing else of note.

Who is this for? Classical Fourier analysts working on summability of subsequences; anyone interested in Zalcwasser's old problems. It deserves a serious referee. The theorem is significant within its area, and the proof, once Lemma 2 is expanded, is complete. Recommendation: send to peer review. I would also cite it if I were working in this corner of harmonic analysis.

Referee Report

0 major / 4 minor

Summary. The paper proves Theorem 1: for 0<α≤1, for every integer-valued polynomial P∈Q[x] of degree r≥2 with positive leading coefficient, and every strictly increasing sequence a_k=P(k) eventually, the (C,α) means σ^a_{N,α}f(x) of the Fourier partial sums converge to f(x) at every Lebesgue point of every f∈L^1(T). This gives affirmative answers to both Zalcwasser questions, in a stronger pointwise form. The proof has three parts: a kernel representation of σ^a_{N,α}f(x)-f(x) as an integral against φ_x(t), a polynomial Weyl estimate (Lemma 2) bounding exponential sums uniformly in lower-order coefficients, and a general Lebesgue-point convergence criterion (Lemma 3) based on a monotone majorant of the kernel satisfying an integrability condition.

Significance. If correct, the result is a genuine advance: it resolves a classical problem from 1936 and improves the a.e. conclusion to convergence at every Lebesgue point. The proof is concise and uses standard tools (van der Corput exponential-sum estimates, Riemann–Lebesgue lemma, Toeplitz summation). The main potential vulnerability, the uniformity in Lemma 2, is handled correctly: the application to u↦P(N−u)t with lower-order coefficients depending on N is legitimate because the r-th derivative depends only on the leading coefficient. The paper is well within the scope of math.CA and should be of interest to researchers in Fourier analysis and exponential sums.

minor comments (4)

[Abstract and §1] The abstract and the first paragraph of the Introduction state that P is 'of degree with positive leading coefficient'; the degree r≥2 is omitted and should be inserted.
[Lemma 2 proof] The proof of Lemma 2 is compressed; please state the exact van der Corput r-th derivative estimate from [5] being used, and spell out the case split, including the regime where t is bounded away from zero, so that the uniformity in the lower-order coefficients and the role of η0 are transparent.
[§2, equation (2.10)] The equality in the last line of (2.10) is correct but suppresses the cancellation of the δ-boundary term; a one-line explanation would improve readability.
[§2, constants] The symbol C is reused for many different constants; for clarity, consider using C(c,r,η) in Lemma 2 and C_α,r,θ in the proof of Theorem 1, or explicitly state that constants may change from line to line.

Circularity Check

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No significant circularity: the proof is a self-contained derivation from standard external results and classical identities.

full rationale

The paper's central claim, Theorem 1, is proved by reducing the convergence of the (C,alpha) means to a kernel estimate, which in turn is derived from a polynomial Weyl estimate. Lemma 2 is quoted from the standard van der Corput theory (Graham and Kolesnik) and is not imported from the author's own prior work; no self-citation is load-bearing. The proof of Theorem 1 does not assume the convergence that it proves. The weighted exponential-sum estimate (2.14) is established from Lemma 2 via Abel summation, and the majorant M_N in (2.18) is then explicitly checked against the boundedness condition (2.6) in Lemma 3. No fitted parameter is renamed as a prediction, no known empirical pattern is relabeled as a derivation, and no uniqueness theorem by the same authors is invoked to force a choice. The Lebesgue-point condition is used in the standard way through the estimate (2.7), and the Riemann-Lebesgue and Toeplitz arguments in Lemma 3 are classical. The paper is therefore not circular; it is a straightforward, if technically delicate, deduction from external standard results.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The ledger is light. The proof relies on standard Fourier identities and on three classical external tools: van der Corput's derivative estimate (cited to [5]), Riemann-Lebesgue, and Toeplitz summation. The only hand-chosen quantity is the auxiliary exponent θ, which is used in the Abel summation step and is not fitted to any data. No new entities, forces, or physical constants are introduced.

free parameters (1)

θ = any value in (0, min{η0(r), α/r}); existence only
Auxiliary exponent introduced in the proof of Theorem 1 (equation (2.13)); it makes the Abel summation exponents α+β>1 and keeps the majorant M_N finite. The theorem is independent of which admissible θ is used, and θ is not fitted to data.

assumptions (4)

standard math van der Corput r-th derivative estimate for exponential sums, as stated in [5]
Used inside Lemma 2 to prove the polynomial Weyl estimate (2.4); the exact form and uniformity constants are imported from [5].
standard math Riemann-Lebesgue lemma
Used in Lemma 3 to show c_k = ∫δ^π φ_x(t) sin((a_k+1/2)t)/(2 sin(t/2)) dt tends to 0.
standard math Toeplitz summation lemma
Used in Lemma 3 to pass from c_k→0 and λ_{N,k}→0 for fixed k with sum_k λ_{N,k}=1 to convergence of the far-tail weighted average.
standard math Cesàro identities (2.1) and asymptotic (2.2) for A_β^n
Used to normalize the (C,α) kernel and to estimate λ_{N,k} in Lemma 3.

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Cite this review

Pith. "Pith review of Cesaro Means along Polynomial Subsequences of Fourier Partial Sums at Lebesgue Points." pith.science (2026). https://pith.science/paper/75H5NSHV

@misc{pith2026260814008,
  author       = {Pith},
  title        = {Pith review of: Cesaro Means along Polynomial Subsequences of Fourier Partial Sums at Lebesgue Points},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/75H5NSHV}},
  note         = {Machine review of arXiv:2608.14008}
}

abstract

In 1936, Zalcwasser proved the almost everywhere convergence of the arithmetic means of the square subsequence of trigonometric Fourier partial sums and asked whether this result extends to higher powers and to Ces\`aro means of fractional order. We give affirmative answers to both questions in a stronger pointwise form. Let $0<\alpha\leq 1$, and let $P$ be an integer-valued polynomial of degree with positive leading coefficient. We prove that the $(C,\alpha)$ means of the Fourier partial sums along any sequence whose $k$-th term equals $P(k)$ for all sufficiently large $k$ converges to $f(x)$ at every Lebesgue point $x$ of every $f\in L^{1}(\T)$.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

A Counterexample to Belinsky's Conjecture on Ces\`aro Means at Lebesgue Points
math.CA 2026-08 reject novelty 7.0 of 10

The paper asserts an explicit convex sequence with a_m <= 7m^8 and an L1 function whose Cesaro Fourier means are unbounded at a Lebesgue point, which would disprove Belinsky's conjecture.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages · cited by 1 Pith paper

[5]

S. W. Graham and G. Kolesnik,Van der Corput’s Method of Exponential Sums, London Mathematical Society Lecture Note Series, vol. 126, Cambridge University Press, Cambridge, 1991

work page 1991
[1]

E. S. Belinsky, Summability of Fourier series with the method of lacunary arithmetical means at the Lebesgue points,Proc. Amer. Math. Soc.125(1997), no. 12, 3689–3693

work page 1997
[2]

E. S. Belinsky, On the summability of Fourier series with the method of lacunary arithmetic means,Anal. Math. 10(1984), no. 4, 275–282

work page 1984
[3]

Gát, Cesàro means of subsequences of partial sums of trigonometric Fourier series,Constr

G. Gát, Cesàro means of subsequences of partial sums of trigonometric Fourier series,Constr. Approx.49(2019), no. 1, 59–101

work page 2019
[4]

Gát, Almost everywhere divergence of Cesàro means of subsequences of partial sums of trigonometric Fourier series,Math

G. Gát, Almost everywhere divergence of Cesàro means of subsequences of partial sums of trigonometric Fourier series,Math. Ann.389(2024), no. 4, 4199–4231

work page 2024
[6]

Izumi and T

S. Izumi and T. Kawata,Notes on Fourier Series, (X). Summability, Tohoku Math. J. First Series46(1940), 154–158

work page 1940
[7]

Lebesgue,Recherches sur la convergence des séries de Fourier,Math

H. Lebesgue,Recherches sur la convergence des séries de Fourier,Math. Ann.61(1905), 251–280

work page 1905
[8]

Zalcwasser,Sur la sommabilité des séries de Fourier,Studia Math.6(1936), 82–88

Z. Zalcwasser,Sur la sommabilité des séries de Fourier,Studia Math.6(1936), 82–88

work page 1936

Show all 9 references

[9]

Zygmund,Trigonometric Series, 2nd ed., vols

A. Zygmund,Trigonometric Series, 2nd ed., vols. I and II, Cambridge University Press, Cambridge, 1959. U. Goginava, Department of Mathematical Sciences, United Arab Emirates University, P.O. Box No. 15551, Al Ain, Abu Dhabi, UAE Email address:zazagoginava@gmail.com; ugoginava@...

1959

Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.

[1] [5]

S. W. Graham and G. Kolesnik,Van der Corput’s Method of Exponential Sums, London Mathematical Society Lecture Note Series, vol. 126, Cambridge University Press, Cambridge, 1991

work page 1991

[2] [1]

E. S. Belinsky, Summability of Fourier series with the method of lacunary arithmetical means at the Lebesgue points,Proc. Amer. Math. Soc.125(1997), no. 12, 3689–3693

work page 1997

[3] [2]

E. S. Belinsky, On the summability of Fourier series with the method of lacunary arithmetic means,Anal. Math. 10(1984), no. 4, 275–282

work page 1984

[4] [3]

Gát, Cesàro means of subsequences of partial sums of trigonometric Fourier series,Constr

G. Gát, Cesàro means of subsequences of partial sums of trigonometric Fourier series,Constr. Approx.49(2019), no. 1, 59–101

work page 2019

[5] [4]

Gát, Almost everywhere divergence of Cesàro means of subsequences of partial sums of trigonometric Fourier series,Math

G. Gát, Almost everywhere divergence of Cesàro means of subsequences of partial sums of trigonometric Fourier series,Math. Ann.389(2024), no. 4, 4199–4231

work page 2024

[6] [6]

Izumi and T

S. Izumi and T. Kawata,Notes on Fourier Series, (X). Summability, Tohoku Math. J. First Series46(1940), 154–158

work page 1940

[7] [7]

Lebesgue,Recherches sur la convergence des séries de Fourier,Math

H. Lebesgue,Recherches sur la convergence des séries de Fourier,Math. Ann.61(1905), 251–280

work page 1905

[8] [8]

Zalcwasser,Sur la sommabilité des séries de Fourier,Studia Math.6(1936), 82–88

Z. Zalcwasser,Sur la sommabilité des séries de Fourier,Studia Math.6(1936), 82–88

work page 1936