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arxiv: 2606.02299 · v1 · pith:22FKBH37 · submitted 2026-06-01 · math.CA

Sharp sign uncertainty for trigonometric polynomials

Tolibjon Ismoilov This is my paper

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Figure 1
Figure 1. Figure 1: Maximizers for the problem ρ(dx, N) when dx is the Lebesgue measure. 1.6.2. Uniform measures on higher-dimensional spheres. Let σd be the normalized surface measure on S d , and by Proposition 7 we know its polar part is the measure µd−1 given by dµd−1(θ) = Cd|sin(2πθ)| d−1 dθ. (1.8) As a… reproduced from arXiv: 2606.02299
classification math.CA
keywords trigonometric polynomialssign uncertainty principlessymmetric Borel measuresunit circlesign changesorthogonal polynomialshigher-dimensional spheres
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0 comments X

The pith

For any symmetric Borel measure on the unit circle, the smallest radius of the last sign change is determined for trigonometric polynomials of degree N with non-positive integral.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes sign uncertainty principles by determining, for each symmetric Borel measure μ on the unit circle, the smallest possible radius at which the last sign change can occur in a trigonometric polynomial of exact degree N whose integral against μ is non-positive. This supplies a sharp location bound on oscillations under the integral constraint. The work extends the same determination to polar measures on spheres S^d by showing that the extremal radius reduces exactly to the one-dimensional case through the polar part of the measure. An analogous sharp radius is also obtained for a polynomial version of the problem on the interval [0,1] that uses orthogonal polynomials.

Core claim

For each symmetric Borel measure μ on the unit circle, the smallest radius of the last sign change is determined for trigonometric polynomials of degree N with non-positive μ-integral. The extremal problem on higher-dimensional spheres reduces to the one-dimensional case via the polar part of the measure, and a parallel sharp result holds for orthogonal polynomials on [0,1].

What carries the argument

The symmetric Borel measure μ on the unit circle, which fixes the non-positive integral condition and yields the explicit minimal radius of the last sign change for degree-N trigonometric polynomials.

Load-bearing premise

The measure is symmetric and Borel on the unit circle, the trigonometric polynomials have exact degree N, and the reduction to one dimension via the polar part of the measure holds.

What would settle it

Construct a trigonometric polynomial of exact degree N whose integral against a chosen symmetric measure μ is non-positive yet whose last sign change occurs at a strictly smaller radius than the value determined by the paper.

read the original abstract

We study sign uncertainty principles for trigonometric polynomials of prescribed degree $N$ with respect to a symmetric Borel measure $\mu$ on the unit circle $\mathbb{R}/\mathbb{Z}$. For each such measure, we determine the smallest radius of the last sign change for trigonometric polynomials with non-positive $\mu$-integral. We further extend these results to polar measures on higher-dimensional spheres $\mathbb{S}^d$, showing that the extremal problem reduces to the one-dimensional case via the polar part of the measure, and we establish a polynomial analogue on $[0,1]$ using orthogonal polynomials on the real line.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper claims to determine, for each symmetric Borel measure μ on the unit circle, the smallest radius of the last sign change for trigonometric polynomials of exact degree N having non-positive μ-integral. It extends the result to polar measures on spheres S^d by reduction to the one-dimensional case via the polar part of the measure, and establishes an analogous result for orthogonal polynomials on [0,1].

Significance. If the explicit determinations hold, the work supplies sharp constants in sign uncertainty principles for trigonometric polynomials, a contribution to harmonic analysis. The reduction via polar parts is a standard device, and the parameter-free character of the radius for general μ is a strength.

minor comments (2)
  1. [Introduction] The introduction could state the explicit form of the extremal radius more prominently rather than deferring all formulas to later sections.
  2. Notation for the last sign-change radius (e.g., r_N(μ)) should be introduced once and used consistently; occasional redefinition risks confusion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the supportive summary, recognition of the significance of the sharp constants, and recommendation of minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central claim is a direct determination, for each symmetric Borel measure μ, of the smallest radius of the last sign change for degree-N trigonometric polynomials having non-positive μ-integral; the higher-dimensional reduction to the circle via the polar part is presented as a standard device rather than a fitted or self-referential step. No equations, definitions, or cited results in the abstract reduce the claimed extremal radius to a parameter fitted from the same data, a self-citation chain, or an ansatz smuggled from prior work by the same authors. The derivation is therefore self-contained against external mathematical benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard properties of trigonometric polynomials, Borel measures, and orthogonal polynomials; no free parameters, invented entities, or ad-hoc axioms are visible from the abstract.

axioms (2)
  • standard math Trigonometric polynomials of degree N are finite linear combinations of 1, cos(2π k x), sin(2π k x) for k ≤ N.
    Invoked implicitly when the degree-N condition is used to bound sign changes.
  • standard math Symmetric Borel measures on the circle admit a well-defined integral against continuous functions.
    Required for the non-positive μ-integral condition.

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Works this paper leans on

27 extracted references · 1 canonical work pages

  1. [1]

    Ambrosio, N

    L. Ambrosio, N. Gigli, and G. Savar´ e. Gradient flows in metric spaces and in the space of probabilit y measures. Basel: Birkh¨ auser, 2nd ed. edition, 2008

    2008

  2. [2]

    N. N. Andreev and V. A. Yudin. Positive values of harmonic polynomials. In Function spaces, approximations, and differential equations. Collected papers dedicated to O leg Vladimirovich Besov on his 70th birthday. Transl. from the Russian , pages 39–45. Moscow: Maik Nauka/Interperiodika, 2003

    2003

  3. [3]

    A. G. Babenko. An extremal problem for polynomials. Mat. Zametki , 35(3):181–186, Mar. 1984

    1984

  4. [4]

    Bourgain, L

    J. Bourgain, L. Clozel, and J.-P. Kahane. Principe d’Hei senberg et fonctions positives. Ann. Inst. Fourier , 60(4):1215–1232, 2010

    2010

  5. [5]

    Carneiro, T

    E. Carneiro, T. Ismoilov, and A. P. Ramos. Sign uncertain ty and de Branges spaces, Aug. 2024. arXiv:2408.01186, to appear in Ann. Inst. Fourier

    work page arXiv 2024

  6. [6]

    Carneiro and F

    E. Carneiro and F. Littmann. Extremal functions in de Bra nges and Euclidean spaces. Advances in Mathemat- ics, 260:281–349, Aug. 2014

    2014

  7. [7]

    Carneiro and E

    E. Carneiro and E. Quesada-Herrera. Generalized sign Fo urier uncertainty. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) , 24(3):1671–1704, 2023

    2023

  8. [8]

    Cohn and F

    H. Cohn and F. Gon¸ calves. An optimal uncertainty princi ple in twelve dimensions via modular forms. Invent. Math., 217(3):799–831, 2019

    2019

  9. [9]

    de Branges

    L. de Branges. Hilbert spaces of entire functions. Prent ice-Hall Series in Modern Analysis. Englewood Cliffs, N.J.: Prentice-Hall. ix, 326 p. 102 s. 6d. (1968)., 1968

    1968

  10. [10]

    G. B. Folland and A. Sitaram. The uncertainty principle : A mathematical survey. J. Fourier Anal. Appl. , 3(3):207–238, 1997

    1997

  11. [11]

    Gon¸ calves, D

    F. Gon¸ calves, D. Oliveira e Silva, and J. P. G. Ramos. On regularity and mass concentration phenomena for the sign uncertainty principle. J. Geom. Anal. , 31(6):6080–6101, 2021

    2021

  12. [12]

    Gon¸ calves, D

    F. Gon¸ calves, D. Oliveira e Silva, and J. P. G. Ramos. Ne w Sign Uncertainty Principles. Discrete Anal. , July 2023

    2023

  13. [13]

    Gon¸ calves, D

    F. Gon¸ calves, D. Oliveira e Silva, and S. Steinerberge r. Hermite polynomials, linear flows on the torus, and an uncertainty principle for roots. J. Math. Anal. Appl. , 451(2):678–711, 2017

    2017

  14. [14]

    M. E. H. Ismail. Classical and quantum orthogonal polynomials in one variab le, volume 98 of Encycl. Math. Appl. Cambridge: Cambridge University Press, 2005

    2005

  15. [15]

    W. B. Jones, O. Nj ˚ astad, and W. J. Thron. Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle. Bull. Lond. Math. Soc. , 21(2):113–152, 1989

    1989

  16. [16]

    X.-J. Li. The Riemann hypothesis for polynomials ortho gonal on the unit circle. Mathematische Nachrichten , 166(1):229–258, Jan. 1994. SHARP SIGN UNCERTAINTY FOR TRIGONOMETRIC POLYNOMIALS 27

    1994

  17. [17]

    X.-J. Li. On reproducing kernel Hilbert spaces of polyn omials. Mathematische Nachrichten , 185(1):115–148, Jan. 1997

    1997

  18. [18]

    Li and D

    X.-J. Li and D. Vaaler. Some trigonometric extremal fun ctions and the Erdos-Turan type inequalities. Indiana University Mathematics Journal , 48(1):0–0, 1999

    1999

  19. [19]

    P. Mattila. Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability , volume 44 of Camb. Stud. Adv. Math. Cambridge: Univ. Press, 1995

    1995

  20. [20]

    B. Simon. Orthogonal Polynomials on the Unit Circle , volume 54.1 of Colloquium Publications . American Mathematical Society, Providence, R.I, 2005

    2005

  21. [21]

    E. M. Stein and G. W eiss. Introduction to Fourier analysis on Euclidean spaces , volume No. 32 of Princeton Mathematical Series . Princeton University Press, Princeton, NJ, 1971

    1971

  22. [22]

    G. Szeg¨ o. ¨Uber die Entwicklung einer analytischen funktion nach den p olynomen eines orthogonalsystems. Math. Ann. , 82:188–212, 1921

    1921

  23. [23]

    G. Szeg¨ o. Orthogonal Polynomials . Number 23 in Colloquium Publications. American mathemati cal society, New York city, online-ausg edition, 1939

    1939

  24. [24]

    M. S. Viazovska. The sphere packing problem in dimensio n 8. Ann. Math. (2) , 185(3):991–1015, 2017

    2017

  25. [25]

    V. A. Yudin. Two external problems for trigonometric po lynomials. Sbornik: Mathematics , 187(11):1721–1736, Dec. 1996

    1996

  26. [26]

    V. A. Yudin. On positive values of spherical harmonics a nd trigonometric polynomials. Math. Notes , 75(3):447– 450, 2004

    2004

  27. [27]

    A. Zygmund. Trigonometric series. Volumes I and II combined. Camb. Math. Libr. Cambridge University Press, 2nd edition, 1988. SISSA - Scuola Internazionale Superiore di Studi A vanzati, Via Bonomea 265, 34136 Trieste, Italy Email address : tolibjon.ismoilov@sissa.it Email address : tolibjon.iismoilov@gmail.com

    1988