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arxiv: 2606.10795 · v1 · pith:7RDGKON2new · submitted 2026-06-09 · 🧮 math.FA

On the refined properties of uniformly quasi-greedy bases

Peiyang Yu This is my paper

Pith reviewed 2026-06-27 11:32 UTC · model grok-4.3

classification 🧮 math.FA
keywords uniformly quasi-greedy basesunconditionalitybounded quotient scalingsstrictly monotone normdisjoint supportsgreedy sumsquasi-greedy constant
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The pith

Unconditionality suffices to keep uniformly quasi-greedy bases stable under bounded-quotient scalings.

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

This paper investigates finer properties of uniformly quasi-greedy bases in normed spaces. It establishes that if a basis is unconditional, then its uniformly quasi-greedy character persists after scaling the basis vectors by a sequence whose ratios are bounded above and below. When the quasi-greedy constant equals one and the norm is strictly monotone, the basis elements must have disjoint supports. The authors also define a strengthened form of quasi-greediness that demands the greedy partial sums remain uniformly bounded in order for any choice of greedy ordering, and derive its basic features.

Core claim

The central results are that unconditionality guarantees stability of uniformly quasi-greediness under scalings with bounded quotient. In the isometric case, 1-uniformly quasi-greediness implies disjointness provided the norm is strictly monotone. A stronger notion requiring uniform order boundedness of the greedy sums is introduced and its properties are characterized similarly to the standard case.

What carries the argument

The uniformly quasi-greedy constant, which quantifies the uniform performance of the greedy algorithm on the basis.

If this is right

  • Any unconditional uniformly quasi-greedy basis remains uniformly quasi-greedy after bounded quotient scalings.
  • 1-uniformly quasi-greedy bases in strictly monotone norms are disjoint.
  • The stronger uniform order boundedness property admits similar stability and characterization results.
  • These properties refine the understanding of how greedy bases behave under perturbations and in special norms.

Where Pith is reading between the lines

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

  • The stability result suggests that unconditional bases provide a robust class for constructing quasi-greedy systems via scaling.
  • Disjointness in the isometric case may link to uniqueness of expansions in spaces with strictly monotone norms.
  • Exploring the stronger notion in concrete Banach spaces like c_0 or l_p could yield new examples.
  • Without strict monotonicity the disjointness may fail, pointing to a separation of properties.

Load-bearing premise

The norm must be strictly monotone to conclude that 1-uniform quasi-greediness forces the basis to be disjoint.

What would settle it

Finding an unconditional basis that is uniformly quasi-greedy but becomes non-quasi-greedy after scaling by a bounded-quotient sequence, or a 1-uniformly quasi-greedy basis with overlapping supports in a strictly monotone norm.

read the original abstract

In this paper, we study some refined properties of uniformly quasi-greedy bases. In particular, we characterize the stability of uniformly quasi-greediness under scalings with bounded quotient and show that unconditionality is sufficient to guarantee this property. For the "isometric" case, i.e., when the uniformly quasi-greedy constant is $1$, we prove that 1-uniformly quasi-greediness implies disjointness if the underlying norm is strictly monotone. We also introduce a stronger notion of quasi-greediness by requiring uniform order boundedness of the greedy sums over greedy orderings. The characterization and properties of such bases are derived along the lines of uniformly quasi-greedy bases.

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 characterizes the stability of uniformly quasi-greediness under scalings with bounded quotient, proving that unconditionality suffices to guarantee this stability. For the isometric case (uniformly quasi-greedy constant equal to 1), it shows that 1-uniformly quasi-greediness implies disjointness provided the underlying norm is strictly monotone. The authors also introduce a stronger notion of quasi-greediness requiring uniform order boundedness of the greedy sums over all greedy orderings and derive its characterization and properties along the same lines as for uniformly quasi-greedy bases.

Significance. If the derivations hold, the results supply concrete stability criteria and a clean implication to disjointness in the isometric setting under a standard monotonicity assumption. The new stronger notion of quasi-greediness enlarges the hierarchy of greedy-type properties with a natural uniform-order condition. These are incremental but technically useful additions to the literature on greedy bases in Banach spaces, obtained from standard definitions without free parameters or ad-hoc constructions.

minor comments (2)
  1. The abstract refers to 'the characterization and properties' of the stronger notion but does not indicate the precise section where the main theorem appears; adding an explicit theorem number in the abstract would improve readability.
  2. Notation for the uniformly quasi-greedy constant (denoted 1 in the isometric case) should be introduced once in §1 or §2 with a forward reference to the stability result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the paper and the recommendation of minor revision. No specific major comments appear in the report, so there are no individual points requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivations characterize stability of uniformly quasi-greediness under bounded-quotient scalings by invoking unconditionality as a sufficient condition, and establish the isometric-case implication to disjointness only under the explicit additional assumption of strict monotonicity of the norm. These steps proceed from the standard definitions of quasi-greediness and unconditionality without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. The stronger notion of quasi-greediness is introduced by definition and its properties are derived along the same lines, remaining self-contained against external benchmarks in greedy basis theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard axioms from Banach space theory and prior definitions of quasi-greedy bases. No free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Standard properties of norms and bases in Banach spaces, including unconditionality and strict monotonicity.
    Invoked for the stability and disjointness results.
  • standard math Definitions of uniformly quasi-greedy bases from the existing literature.
    The paper extends these definitions with new properties.

pith-pipeline@v0.9.1-grok · 5635 in / 1305 out tokens · 23255 ms · 2026-06-27T11:32:07.182441+00:00 · methodology

discussion (0)

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Reference graph

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