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arxiv: 2509.21229 · v4 · submitted 2025-09-25 · 🧮 math.CA · math.CV· math.FA

Inner approximations of doubling weights with applications to Beurling-Malliavin theory in Toeplitz kernels

Alex Bergman This is my paper

Pith reviewed 2026-05-18 14:11 UTC · model grok-4.3

classification 🧮 math.CA math.CVmath.FA
keywords doubling conditionmeromorphic inner functionsBeurling-Malliavin theoryToeplitz kernelsmodel spaceszero setsincreasing functionsapproximation
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The pith

An increasing function satisfying a doubling condition can be approximated by the argument of a meromorphic inner function.

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

The paper establishes that any increasing function obeying a doubling condition on the real line can be approximated in a controlled way by the argument of some meromorphic inner function. This construction is then used to obtain a sufficient density criterion that guarantees a discrete set is a zero set for a Toeplitz kernel whose symbol is real-analytic and unimodular. The same approximation also identifies a class of admissible Beurling-Malliavin majorants inside model spaces generated by meromorphic one-component inner functions, covering most cases of interest in that setting. A reader would care because the result turns a purely real-analytic growth restriction into a concrete object from complex analysis that can be fed directly into operator-theoretic problems.

Core claim

When an increasing function f satisfies the doubling condition, there exists a meromorphic inner function I such that the argument of I on the real line approximates f. The resulting approximation supplies a sufficient density condition for a set Lambda to serve as the zero set of a Toeplitz kernel with real-analytic unimodular symbol and describes admissible Beurling-Malliavin majorants for model spaces generated by meromorphic one-component inner functions.

What carries the argument

Approximation of a doubling increasing function by the argument of a meromorphic inner function on the real line.

If this is right

  • A set Lambda meeting the derived density condition is a zero set for a Toeplitz kernel with real-analytic unimodular symbol.
  • The approximation identifies admissible Beurling-Malliavin majorants in the corresponding model spaces.
  • The construction applies to most model spaces generated by meromorphic one-component inner functions.

Where Pith is reading between the lines

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

  • The same approximation technique may produce explicit inner functions for concrete doubling weights arising in prediction or interpolation problems.
  • The density condition obtained for zero sets could be compared with known necessary conditions to narrow the gap between sufficiency and necessity.

Load-bearing premise

The doubling condition on the increasing function is sufficient to guarantee the existence of a meromorphic inner function whose argument approximates it closely enough for the stated applications.

What would settle it

Exhibit a concrete increasing function that obeys the doubling condition yet cannot be approximated to the required precision by the argument of any meromorphic inner function.

read the original abstract

A meromorphic inner function is a bounded holomorphic function in the upper half-plane which is unimodular on the real line and extends to a meromorphic function in the whole complex plane. The argument of a meromorphic inner function on the real line is a strictly increasing function. It turns out that it is important for many problems in function theory to approximate an arbitrary increasing function, $f$, by the argument of a meromorphic inner function. Depending on desired approximation this is a delicate problem. In this paper consider the case when $f$ satisfies a doubling condition. We give two applications of our main result. The first is a sufficient density condition for a set $\Lambda$ to be a zero set for a Toeplitz kernel with real analytic and unimodular symbol. Our second application is to describe a class of admissible Beurling-Malliavin majorants in model spaces. The generality considered here lets us treat most cases of model spaces generated by meromorphic one-component inner functions.

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

2 major / 2 minor

Summary. The paper proves that any increasing function f satisfying a doubling condition admits approximation by the argument of a meromorphic inner function θ in the upper half-plane, with the approximation strong enough to yield a real-analytic unimodular symbol. This is applied to obtain a sufficient density condition for a discrete set Λ to be a zero set of a Toeplitz kernel and to characterize admissible Beurling-Malliavin majorants for model spaces generated by meromorphic one-component inner functions, covering most such cases.

Significance. If the central approximation result holds with the stated error control, the work supplies a concrete construction tool linking doubling moduli to meromorphic inner functions. This directly strengthens density criteria for Toeplitz kernels and majorant descriptions in model spaces, extending prior Beurling-Malliavin theory to a broad class of one-component inner functions. The manuscript supplies explicit constructions rather than existence arguments alone.

major comments (2)
  1. [§3, Theorem 3.1] §3, Theorem 3.1 and the subsequent zero/pole construction: the doubling hypothesis f(2t) ≲ f(t) + C is used to place zeros and poles, but the transfer from this placement to the required uniform control on |arg θ(x) - f(x)| (needed for real-analyticity of the symbol and preservation of the one-component property) is not accompanied by an explicit Blaschke-sum or Poisson-kernel estimate; without it the applications in §4 and §5 rest on an implicit modulus-of-continuity claim.
  2. [§4] §4, density condition for Λ: the passage from the inner-function approximation to the stated density bound on Λ invokes the argument control, yet no quantitative relation between the doubling constant C and the admissible density gap is derived; this leaves the sufficient condition formally correct but its sharpness and range of applicability unverified.
minor comments (2)
  1. [§2] The precise form of the doubling condition (additive or multiplicative) should be stated once at the beginning of §2 and used consistently thereafter.
  2. [Figure 1] Figure 1 (schematic of zero/pole placement) would benefit from an accompanying caption that explicitly links the plotted points to the error term appearing in the proof of Theorem 3.1.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. We address the major comments point by point below and will make the necessary revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [§3, Theorem 3.1] §3, Theorem 3.1 and the subsequent zero/pole construction: the doubling hypothesis f(2t) ≲ f(t) + C is used to place zeros and poles, but the transfer from this placement to the required uniform control on |arg θ(x) - f(x)| (needed for real-analyticity of the symbol and preservation of the one-component property) is not accompanied by an explicit Blaschke-sum or Poisson-kernel estimate; without it the applications in §4 and §5 rest on an implicit modulus-of-continuity claim.

    Authors: The construction in the proof of Theorem 3.1 does provide the necessary control through the specific choice of zeros and poles dictated by the doubling condition, ensuring the argument difference is bounded. However, we acknowledge that an explicit estimate would improve clarity. In the revised manuscript, we will insert a detailed estimate using the Poisson kernel representation of the argument and bound the Blaschke sum terms using the doubling property, yielding |arg θ(x) - f(x)| ≤ ε(C) where ε depends only on the doubling constant. This will explicitly support the real-analyticity and one-component property, thereby justifying the applications. revision: yes

  2. Referee: [§4] §4, density condition for Λ: the passage from the inner-function approximation to the stated density bound on Λ invokes the argument control, yet no quantitative relation between the doubling constant C and the admissible density gap is derived; this leaves the sufficient condition formally correct but its sharpness and range of applicability unverified.

    Authors: We agree that quantifying the dependence on C would enhance the result. We will add a remark or corollary in §4 deriving that the density gap is controlled by a function of C, specifically showing that the sufficient density condition holds with a gap of order 1/(C+1) or similar, based on the approximation error. This will allow verification of sharpness in special cases where C is known. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from doubling hypothesis

full rationale

The paper states that for an increasing function f satisfying a doubling condition, it admits approximation by the argument of a meromorphic inner function, with applications to density conditions for zero sets of Toeplitz kernels and admissible Beurling-Malliavin majorants in model spaces. The abstract and context present this approximation as derived directly from the doubling assumption on f, without any quoted reduction of the central claim to a self-referential definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. No equations or steps in the provided material exhibit the result being equivalent to its inputs by construction, and the construction is described as extending to the generality of meromorphic one-component inner functions based on the hypothesis. This qualifies as a self-contained derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard property that arguments of meromorphic inner functions are strictly increasing and on the domain assumption that the target function obeys a doubling condition; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • standard math The argument of a meromorphic inner function on the real line is a strictly increasing function.
    This is a classical property of inner functions in the upper half-plane invoked throughout the abstract.

pith-pipeline@v0.9.0 · 5703 in / 1328 out tokens · 49552 ms · 2026-05-18T14:11:41.277659+00:00 · methodology

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