arxiv: 2604.20758 · v1 · submitted 2026-04-22 · 🧮 math.CV

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Stability under product and composition for uniform Carleman asymptotic expansions

Gerhard Schindl, Ignacio Miguel-Cantero, Javier Jim\'enez-Garrido, Javier Sanz

Pith reviewed 2026-05-09 22:18 UTC · model grok-4.3

classification 🧮 math.CV

keywords Carleman classesuniform asymptotic expansionsstability under productcompositionRoumieuBeurlingalgebrabilityFaà di Bruno

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The pith

Certain conditions on a sequence of positive numbers ensure that Carleman classes of holomorphic functions with uniform asymptotic expansions are stable under pointwise product and composition.

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

The paper shows that two standard conditions on the sequence M, algebrability and Faà di Bruno, make Carleman classes closed under pointwise multiplication and under composition. This closure holds in both the Roumieu and Beurling versions of the classes for holomorphic functions on sectors of the Riemann surface of the logarithm. The same conditions are necessary for stability in the Roumieu case once the class contains suitable characteristic functions, which the authors construct using classical existence results. Stability under these operations lets one carry out algebraic manipulations while keeping the precise asymptotic control given by M.

Core claim

Imposing the algebrability and Faà di Bruno conditions on the sequence M ensures the desired stability with respect to point-wise product and composition in both the Roumieu and the Beurling settings for Carleman classes of holomorphic functions admitting uniform asymptotic expansions controlled by M. These conditions turn out to be necessary for the corresponding stability in the Roumieu case as long as the existence of suitable characteristic functions is guaranteed within the class, and the construction of such functions is given in detail.

What carries the argument

The sequence M of positive real numbers controlling the remainder estimates in the uniform asymptotic expansion, together with the algebrability condition (ensuring the class forms an algebra) and the Faà di Bruno condition (controlling derivative growth under composition).

If this is right

The pointwise product of any two functions in the class again belongs to the class and admits an asymptotic expansion controlled by the same sequence M.
The composition of two functions in the class remains in the class with the same controlling sequence M.
In the Roumieu setting, if the conditions on M fail, then stability under product or composition can be lost once characteristic functions are present.
The results extend earlier partial statements for Gevrey classes of order 1 to general Carleman classes in both settings.

Where Pith is reading between the lines

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

The explicit construction of characteristic functions may allow similar stability results to be checked for other operations such as differentiation within the same classes.
One could test whether the necessity of the conditions persists in classes defined on different domains or without the holomorphic assumption.
The sufficiency part might combine with existing approximation theorems to produce new examples of functions with prescribed asymptotics that remain closed under algebraic operations.

Load-bearing premise

The existence of suitable characteristic functions in a precise sense within the Roumieu class is required to prove that the conditions on M are necessary for stability.

What would settle it

A sequence M that satisfies algebrability and Faà di Bruno but for which there exist two functions in the corresponding Carleman class whose pointwise product fails to admit a uniform asymptotic expansion controlled by M would falsify the sufficiency claim.

read the original abstract

We study the stability under point-wise product and under composition in Carleman classes of holomorphic functions, defined on sectors of the Riemann surface of the logarithm, and admitting a uniform asymptotic expansion with remainders controlled by a given sequence of positive real numbers $\mathbf{M}$. On the one hand, the well-known conditions of algebrability and Fa\`a di Bruno, imposed on the sequence $\mathbf{M}$, ensure the desired stability with respect to each operation in both the Roumieu and the Beurling settings. On the other hand, these conditions turn out to be necessary for the corresponding stability in the Roumieu case as long as the existence of suitable characteristic functions, in a precise sense, is guaranteed within the class. The construction of such functions rests on classical results of B. Rodr\'iguez-Salinas, and is given in detail. Our results are inspired by, and thoroughly generalize, several partial statements by G.~Auberson and G.~Mennessier for Gevrey classes of order 1.

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.

Desk Editor's Note private letter to a colleague

This paper extends stability under product and composition from Gevrey order 1 to general uniform Carleman classes, with new necessity results in the Roumieu case when characteristic functions exist.

read the letter

The core contribution is a clean generalization: the algebrability and Faà di Bruno conditions on the sequence M suffice for stability under pointwise product and composition in both Roumieu and Beurling settings for uniform Carleman expansions on sectors. The authors also show necessity in the Roumieu case provided suitable characteristic functions live in the class, and they supply an explicit construction drawing on Rodríguez-Salinas. This moves past the partial Auberson-Mennessier statements for Gevrey order 1 and fills in the necessity direction with concrete details. The sufficiency argument rests on standard conditions that are already known to work in narrower settings, so that part feels solid and unsurprising once stated. The necessity part is the fresher piece because it is conditional but made usable by the construction they spell out. No circularity appears; the logic flows from the classical results they cite. One soft spot is the conditional nature of necessity: if the characteristic functions are absent from a given class, the necessity claim does not apply. The paper is upfront about this and provides the construction for many cases, so the limitation is transparent rather than hidden. The overall structure is coherent and the citation pattern draws appropriately on the relevant classical sources without excess self-reference. This work is aimed at specialists already working with Carleman classes, uniform asymptotics, or holomorphic functions on Riemann surfaces. A reader who knows the Gevrey case will see the value in the extension and the explicit necessity argument. It has enough formal grounding and new content to merit a serious referee, even if the proofs need polishing for readability.

Referee Report

1 major / 3 minor

Summary. The paper examines stability under pointwise product and composition for holomorphic functions on sectors of the Riemann surface of the logarithm that admit uniform Carleman asymptotic expansions controlled by a sequence M. It proves that the algebrability and Faà di Bruno conditions on M suffice for stability in both the Roumieu and Beurling settings. Necessity of these conditions holds in the Roumieu case conditional on the existence of suitable characteristic functions in the class, with an explicit construction of such functions provided via classical results of Rodríguez-Salinas. The results generalize partial statements of Auberson and Mennessier for Gevrey classes of order 1.

Significance. If the results hold, this work supplies a nearly complete characterization of sequences M for which Carleman classes are stable under product and composition, extending beyond the Gevrey case with explicit constructions that address the necessity direction. The unconditional sufficiency in both Roumieu and Beurling settings, combined with the detailed handling of characteristic functions, strengthens the contribution to the theory of asymptotic expansions and holomorphic function classes.

major comments (1)

[Necessity section (around the application of Rodríguez-Salinas results)] The necessity result in the Roumieu setting is correctly framed as conditional on the existence of characteristic functions, but the manuscript should explicitly state (perhaps in the introduction or the necessity section) whether the Rodríguez-Salinas construction guarantees such functions for every M satisfying algebrability and Faà di Bruno, or if additional restrictions on M or the sector apply; this would clarify the scope of the necessity claim.

minor comments (3)

[Abstract and Introduction] The abstract refers to 'sectors of the Riemann surface of the logarithm' without specifying the opening angle or the precise definition of the Carleman class; a brief reminder in the introduction would aid readers unfamiliar with the uniform asymptotic setting.
[Preliminaries] Notation for the sequence M and the associated Carleman classes (Roumieu vs. Beurling) should be introduced with a dedicated preliminary subsection to avoid scattered definitions later in the text.
[Introduction] The generalization of Auberson-Mennessier statements is claimed to be thorough, but a short table or explicit list comparing the new results to the original Gevrey-order-1 statements would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and the recommendation for minor revision. We address the single major comment below and will incorporate the suggested clarification.

read point-by-point responses

Referee: [Necessity section (around the application of Rodríguez-Salinas results)] The necessity result in the Roumieu setting is correctly framed as conditional on the existence of characteristic functions, but the manuscript should explicitly state (perhaps in the introduction or the necessity section) whether the Rodríguez-Salinas construction guarantees such functions for every M satisfying algebrability and Faà di Bruno, or if additional restrictions on M or the sector apply; this would clarify the scope of the necessity claim.

Authors: We agree that an explicit statement on the scope would improve clarity. The construction via Rodríguez-Salinas applies to every sequence M satisfying the algebrability and Faà di Bruno conditions, with no additional restrictions on M or the sector (beyond the standard assumptions for the Carleman classes on sectors of the Riemann surface of the logarithm). We will add a clarifying sentence to this effect in the introduction and in the necessity section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via external classical results

full rationale

The paper's central claims rest on sufficiency of the standard algebrability and Faà di Bruno conditions on M (well-known and external) for stability in both Roumieu and Beurling settings, plus necessity in Roumieu conditional on existence of characteristic functions whose explicit construction is supplied by classical Rodríguez-Salinas results. No step equates a derived quantity to its own input by definition, renames a fitted parameter as a prediction, or loads the argument on a self-citation chain. The generalization of Auberson-Mennessier statements proceeds from these independent inputs without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the standard mathematical definitions of Carleman classes, Roumieu and Beurling settings, and the classical construction of characteristic functions; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Existence of suitable characteristic functions within the class (in a precise sense)
Invoked to establish necessity of the algebrability and Faà di Bruno conditions in the Roumieu case.

pith-pipeline@v0.9.0 · 5485 in / 1255 out tokens · 48304 ms · 2026-05-09T22:18:57.089840+00:00 · methodology

discussion (0)

Reference graph

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