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arxiv: 2606.04640 · v1 · pith:MH2K5CPCnew · submitted 2026-06-03 · 🧮 math.CA

From local to global asymptotic behaviour of orthogonal polynomials

Roman Bessonov , Artur Nicolau This is my paper

Pith reviewed 2026-06-28 03:59 UTC · model grok-4.3

classification 🧮 math.CA
keywords orthogonal polynomialsunit circleSzegő classCesàro asymptoticsStolz anglesreflected polynomialsreproducing kernels
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The pith

For Szegő-class measures, the Cesàro means of |reflected orthogonal polynomial times Szegő function| squared converge uniformly to 1 inside almost every Stolz angle.

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

The paper proves that for orthogonality measures in the Szegő class, the average over the first n reflected polynomials of the squared modulus |ϕ_k^*(z) D_μ(z)|^2 approaches 1 uniformly for z inside almost every Stolz angle Γ_ζ at the boundary. This moves the known pointwise local asymptotics of Máté-Nevai-Totik, which operate at distance scale 1/n near the circle, to a global statement that holds at scale 1 throughout the disk. The work also derives new results on the arguments of the polynomials that extend the Grenander-Szegő theorem and obtains global asymptotics for the associated reproducing kernels under varied assumptions on μ.

Core claim

Let {ϕ_n^*} be the sequence of reflected orthogonal polynomials on the unit circle ∂D generated by a measure μ of Szegő class, and let D_μ be the Szegő function of μ. We prove the uniform Cesàro asymptotics sup_{z ∈ Γ_ζ} (1/n ∑_{k=0}^{n-1} |ϕ_k^*(z) D_μ(z)|^2 - 1|) → 0 as n→∞ for almost all Stolz angles Γ_ζ, ζ∈∂D. This extends a well-known asymptotic result of Máté, Nevai, and Totik (1991) from the local scale O(1/n) near ∂D to the global scale O(1). We also study asymptotic behavior of arguments of orthogonal polynomials and extend a classical theorem due to Grenander and Szegő using a new technique. As an application, we derive global asymptotic results for polynomial reproducing kernels u

What carries the argument

The uniform Cesàro mean (1/n)∑_{k=0}^{n-1} |ϕ_k^*(z) D_μ(z)|^2 taken inside Stolz angles Γ_ζ for almost every boundary point ζ, which upgrades local boundary control to global interior control.

If this is right

  • The local Máté-Nevai-Totik asymptotics extend to uniform global convergence inside almost every Stolz angle.
  • Asymptotic results on the arguments of the orthogonal polynomials extend the classical Grenander-Szegő theorem.
  • Global asymptotic formulas hold for the polynomial reproducing kernels under varied assumptions on the measure μ.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The averaging technique may allow passage from local to global statements in related families of orthogonal polynomials on other contours.
  • The result suggests that local boundary behavior of the measure controls averaged interior quantities for almost every approach direction.

Load-bearing premise

The orthogonality measure μ belongs to the Szegő class so that the Szegő function D_μ is well-defined and the local Máté-Nevai-Totik asymptotics apply.

What would settle it

A Szegő-class measure μ together with a positive-measure set of boundary points ζ where the Cesàro average of |ϕ_k^*(z) D_μ(z)|^2 stays bounded away from 1 uniformly inside the Stolz angle Γ_ζ for infinitely many n.

Figures

Figures reproduced from arXiv: 2606.04640 by Artur Nicolau, Roman Bessonov.

Figure 1
Figure 1. Figure 1: Different asymptotic methods work in different regions. On compact subsets of D, the classical Szegő argument applies. Near the unit circle, at disks of radii ∼ 1/n, one can use Máté, Nevai, and Totik approach. We prove the uniform Cesàro asymp￾totics in the whole Stolz angle Γζ . See also Theorem 4.8 for the convergence in the bigger domain shown in Figure 1b. Theorem 2.2. Let µ ∈ Sz(T) and ζ ∈ T be such … view at source ↗
Figure 2
Figure 2. Figure 2: Objects appearing in the proof of Lemma 3.2. with constants depending only on A1, A2 and ε. This gives [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The region appearing in Theorem 4.8 (filled with gray) contain the region from Figure 1b (filled here with dots). The theorem follows by letting ε → 0. □ We have proved Theorem 2.1 for Stolz angles. The same method gives a bit more general result that we formulate below. Theorem 4.8. Let µ ∈ Sz(T) and ζ ∈ T be such that (2.1) holds and |Dµ| has a non-zero finite non-tangential limit at ζ. Fix A > 0 and let… view at source ↗
read the original abstract

Let $\{\phi^*_n\}$ be the sequence of reflected orthogonal polynomials on the unit circle $\partial \mathbb{D}$ generated by a measure $\mu$ of Szeg\H{o} class, and let $D_{\mu}$ be the Szeg\H{o} function of $\mu$. We prove the uniform Ces\`aro asymptotics $$ \sup_{z \in \Gamma_\zeta}\Biggl(\frac{1}{n}\sum_{k = 0}^{n-1}\Bigl||\phi_k^*(z) D_{\mu}(z)|^2 - 1\Bigr|\Biggr) \to 0, \qquad n \to \infty, $$ for almost all Stolz angles $\Gamma_{\zeta}$, $\zeta\in \partial \mathbb{D}$. This extends a well-known asymptotic result of M\'at\'e, Nevai, and Totik (1991) from the local scale $O(1/n)$ near $\partial \mathbb{D}$ to the global scale $O(1)$. We also study asymptotic behavior of arguments of orthogonal polynomials and extend a classical theorem due to Grenander and Szeg\H{o} using a new technique. As an application, we derive global asymptotic results for polynomial reproducing kernels under various assumptions on the orthogonality measure.

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 / 3 minor

Summary. The manuscript proves uniform Cesàro asymptotics sup_{z ∈ Γ_ζ} (1/n ∑_{k=0}^{n-1} ||ϕ_k^*(z) D_μ(z)|^2 - 1|) → 0 as n→∞ for almost all Stolz angles Γ_ζ with ζ∈∂D, where {ϕ_n^*} are the reflected orthogonal polynomials on the unit circle generated by a Szegő-class measure μ with Szegő function D_μ. This extends the local-scale Máté-Nevai-Totik asymptotics to the global O(1) scale. The paper also studies argument asymptotics of orthogonal polynomials (extending Grenander-Szegő via a new technique) and derives global asymptotics for polynomial reproducing kernels under various measure assumptions.

Significance. If the central extension holds, the result supplies a global counterpart to the well-known local MNT asymptotics in OPUC theory, which may streamline applications to reproducing kernels. The new technique for the argument asymptotics is a concrete technical contribution.

minor comments (3)
  1. [Introduction] Define the reflected polynomials ϕ_k^* and the Stolz angles Γ_ζ explicitly in the introduction (or §1) rather than assuming familiarity with the notation.
  2. [Applications section] In the application to reproducing kernels, state the precise assumptions on μ (beyond Szegő class) that are used for each global asymptotic statement.
  3. [Main theorem statement] Clarify whether the 'almost all' qualifier for ζ arises from the same null set as in the MNT local result or from an additional argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the accurate summary of its contributions, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central result extends the external Máté-Nevai-Totik (1991) local asymptotics to a uniform Cesàro mean over Stolz angles Γ_ζ for a.e. ζ, under the Szegő-class hypothesis that defines D_μ. This is a standard extension of an independent classical theorem rather than any reduction of the new global statement to a fitted quantity or self-defined input. No self-citation load-bearing steps, self-definitional relations, or ansatz smuggling appear in the provided abstract or description. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the standard definition of Szegő-class measures and the existence of the Szegő function; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption μ is a Szegő-class measure on the unit circle
    Required for D_μ to exist and for the cited local asymptotics to hold.

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