Metadata
bibkey: acostababb2024partial
authors: Ryan L. Acosta Babb
year: 2024
title: A partial converse to the Riemann–Lebesgue lemma for Bessel–Fourier series of order zero
doi: 10.48550/arXiv.2410.17681
url: https://arxiv.org/abs/2410.17681v1
claim: "Conjecture 1 asserts nonnegativity of the explicit weighted order-zero Bessel kernel for 0 <= alpha <= 1/2, every natural mode count and every x in [0,1]."
strata_touched:
- D5/S3/Analytic/BesselRedheffer/AcostaBabbKernelRefutation
license: citation-only
triage: anchor
Verified locator
DOI: 10.48550/arXiv.2410.17681
Source: https://arxiv.org/abs/2410.17681v1
Full text: https://arxiv.org/pdf/2410.17681v1
Source assertion and definitions
Conjecture 1, p. 10, verbatim TeX:
For all, $0\leq\alpha\leq\frac{1}{2}$, $M\in\N$ and $x\in [0,1]$, $K^{\alpha}_M(x)\geq 0$.
The macro \N is \mathbb{N}. Lemma 2.2, p. 6, states:
Let $j_m$ denote the $m\th$ positive zero of $J_0$. Then
Section 3, p. 7, states:
Let us write $$S^{\alpha}_m(x) = \sum_{k=1}^{m}j_m^{\alpha}J_0(j_k x).$$
We will need the following `modified Fej\'er kernel' for the Bessel--Fourier series:
\[
K^{\alpha}_M(x) := \frac{1}{M+1}\sum_{m=1}^{M}S^{\alpha}_m(x)
= \sum_{m=1}^{M}\left(1-\frac{m}{M+1}\right)j_m^{\alpha}J_0(j_m x).
\] We will also set $K^{\alpha}_0(x) \equiv 1$ when $M=0$.
The printed S has a factor indexed by m inside the sum over k; it does
not imply the displayed Cesaro equality. The refutation concerns the
explicit weighted formula on the right and its separate value at M=0,
as specified in #22175.
The printed S is kept separately. The real owner nbessel 0 is the
standard order-zero Bessel series. Positive zeros are indexed from one.
The recursive infimum sequence needs only its first two entries for this
counterexample; their attainment and rational enclosures are proved.
Scope of the refutation
The repository refutes the assertion at alpha=1/2, M=2, x=999/1000. It does not prove the proposed uniform alpha rectangle or sharp M=2 threshold, or decide positivity for other mode counts. The false positivity premise removes the proposed conditional route to the source’s stronger coefficient construction; it does not refute those existence conclusions or Conjecture 2.
The journal version, On the rate of growth of Bessel–Fourier coefficients for integrable functions, J. Math. Anal. Appl. 559 (2026), 130440, DOI 10.1016/j.jmaa.2026.130440, retains Conjecture 1 on p. 11. The target is the unambiguous order-zero arXiv v1 statement above.
Computation and literature boundary
The exact rational experiment is in
the-omega-institute/trureturing-experiments, commit
b01beb85ba641ba44f5d5711499d06a46af0ddef, at
docs/reports/acosta-babb-bessel-kernel/check.py, SHA-256
ce3c873f27837df0248c09ec4c25d95d6a159658aeaed7ef0457db174a8acef1.
The command python3 docs/reports/acosta-babb-bessel-kernel/check.py
has recorded exit code 0 and final output ALL_CHECKS_OK in #22175.
It checks rational inequalities for root signs, the point enclosure,
the endpoint-flux margin 27843/500000, curvature bound
895734319/60000000 and power comparisons for a proposed threshold
between 17/50 and 7/20. The uniform-family and threshold analytic
bridges are not Lean theorems of this delivery.
Literature completeness is ASSUMED-UNVERIFIED. The partial screen in #22175 found no settlement in the texts successfully read. MathOverflow, rate-limited citation queries, unread final journal texts, uncapped broader arXiv results and full classical treatises remain outside that screen. The refutation is repository-derived; no originality claim is made.