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arxiv: 2505.08839 · v2 · submitted 2025-05-13 · 🧮 math.CA

On the equivalence between moderate growth-type conditions in the weight matrix setting II

Gerhard Schindl This is my paper

Pith reviewed 2026-05-22 16:04 UTC · model grok-4.3

classification 🧮 math.CA
keywords moderate growthweight sequencesweight functionsmixed settingweight matrixBraun-Meise-Taylor
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The pith

The moderate growth property for mixed weight sequences admits a new characterization in terms of the associated weight function.

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

The paper extends previous work on equivalent conditions for the moderate growth of weight sequences to the case involving two different sequences. It establishes that a full set of equivalences does not hold in this mixed setting, but a new characterization becomes available when the weight function is constructed from a weight sequence. This is relevant in the broader context of weight matrices and Braun-Meise-Taylor weight functions used to define classes of ultradifferentiable functions. Understanding these equivalences helps in transferring properties between different descriptions of growth conditions in analysis.

Core claim

In the main result, the moderate growth condition in the mixed setting is characterized equivalently by a property of the associated weight function when this function is based on a weight sequence.

What carries the argument

The associated weight function derived from the weight sequence, which encodes the moderate growth property equivalently in the mixed case.

If this is right

  • The equivalence provides a practical way to check moderate growth using the weight function description.
  • It applies specifically in contexts where weight functions arise from sequences.
  • Full generalization of all classical equivalences fails for arbitrary mixed sequences.

Where Pith is reading between the lines

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

  • If the characterization holds, it may simplify proofs involving weight matrices by reducing to function properties.
  • This could extend to other growth conditions in the study of Denjoy-Carleman classes.

Load-bearing premise

The weight function is constructed from a weight sequence.

What would settle it

A pair of weight sequences where the moderate growth condition holds but the corresponding property for the associated weight function does not, or the reverse.

read the original abstract

We continue the study of the known equivalent reformulations of the classical moderate growth condition for weight sequences in the mixed setting; i.e. when dealing with two different sequences. This approach is becoming crucial in the weight matrix setting and also, in particular, when dealing with weight functions in the sense of Braun-Meise-Taylor. It is known that a full generalization to the mixed setting fails, more precisely the condition comparing the growth of the corresponding sequences of quotients and roots is not clear. In the main result we prove a new characterization of this property in terms of the associated weight function; i.e. when the given weight function is based on a weight sequence.

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 paper continues prior work on equivalent reformulations of the classical moderate growth condition for weight sequences, now in the mixed setting with two different sequences. It is known that a full generalization fails; the main result establishes a new characterization of the moderate growth property in terms of the associated weight function, specifically when that weight function is constructed from a weight sequence (in the sense of Braun-Meise-Taylor).

Significance. If the stated characterization holds, the result supplies a concrete, usable reformulation that respects the known obstructions to full mixed-setting generalization. This is a modest but useful incremental advance for the theory of weight matrices and ultradifferentiable classes, particularly when one must pass between sequence-based and function-based descriptions.

minor comments (3)
  1. The abstract and introduction repeatedly refer to 'the associated weight function' without an early, self-contained definition or pointer to the precise construction from the weight sequence; a short preliminary subsection or displayed formula would improve readability.
  2. Notation for the mixed setting (e.g., the two sequences M and N, or the corresponding weight functions) should be fixed once at the beginning and used consistently; occasional switches between subscript and superscript conventions appear in the provided text.
  3. The manuscript cites earlier results on the non-mixed case and on the failure of full generalization, but does not include a brief comparative table or diagram summarizing which equivalences survive and which fail; such a visual aid would help readers situate the new characterization.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our work and the recommendation for minor revision. The assessment that the result provides a concrete reformulation respecting known obstructions in the mixed setting is appreciated.

Circularity Check

0 steps flagged

No significant circularity; new characterization rests on external definitions

full rationale

The paper continues prior work on moderate growth conditions for weight sequences in the mixed setting and explicitly notes that full generalization fails. The main result is a new characterization of the property specifically when the weight function is derived from a weight sequence, using standard external definitions of weight functions and sequences rather than any internal fit or self-referential construction. No load-bearing step reduces by construction to a fitted parameter, renamed input, or unverified self-citation chain; the argument is a proof of equivalence under the stated restriction and remains self-contained against the cited external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper operates within the established framework of weight sequences and weight functions from prior literature on ultradifferentiable classes. No new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption Weight sequences satisfy standard regularity conditions such as log-convexity and moderate growth in the non-mixed case.
    These are background assumptions from the classical theory that the mixed-setting extension builds upon.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. On inclusion relations of weighted $L^p$-type spaces defined in terms of weight function matrices

    math.FA 2025-10 unverdicted novelty 6.0

    New weighted L^p spaces are defined via weight function matrices with inclusion relations characterized by matrix comparisons; a counterexample distinguishes Beurling-Björck from Braun-Meise-Taylor ultradifferentiable...

Reference graph

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