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When Greedy Approximation Breaks: Counterexamples in Quasi-Banach Spaces

Fernando Albiac, José L. Ansorena, Miguel Berasategui, Pablo M. Berná

TL;DR

This work resolves two long-standing open problems in greedy approximation theory within quasi-Banach spaces by producing two counterexamples. First, it shows that an almost greedy basis in a quasi-Banach space can lose quasi-greediness upon passing to its Banach envelope, illustrating that envelope regularization can drastically alter greedy performance. Second, it exhibits an almost greedy Markushevich basis in a nonlocally convex quasi-Banach space that cannot be rearranged into a Schauder basis, underscoring the independence of greedy behavior from Schauder structure. Together, these results reveal that local convexity and envelope constructions critically shape greedy phenomena beyond classical Banach-space theory and highlight structural distinctions between Banach and quasi-Banach contexts.

Abstract

We construct two counterexamples that resolve long-standing open problems on greedy approximation theory with respect to bases, posed in [F. Albiac et al., Dissertationes Math. 560 (2021)] and restated in [F. Albiac, J. L. Ansorena, V. Temlyakov, J. Approx. Theory 307 (2025)]. Our first result exhibits a quasi-Banach space $\mathbb{X}$ with an almost greedy basis which, when transported to the Banach envelope of $\mathbb{X}$, ceases to be quasi-greedy. This shows that the passage to the Banach envelope, although it preserves linear and lattice structure, may radically disrupt the performance of the thresholding greedy algorithm, to the extent that in some respects it could perform better in a quasi-Banach space than in its Banach envelope. Our second result constructs an almost greedy Markushevich basis in a nonlocally convex quasi-Banach space $\mathbb{Y}$ which fails to be a Schauder basis under any reordering. Together, these examples highlight that local convexity and the Banach envelope construction play an unexpectedly active role in shaping greedy approximation phenomena, revealing structural differences between Banach and quasi-Banach spaces that go beyond the classical theory of bases.

When Greedy Approximation Breaks: Counterexamples in Quasi-Banach Spaces

TL;DR

This work resolves two long-standing open problems in greedy approximation theory within quasi-Banach spaces by producing two counterexamples. First, it shows that an almost greedy basis in a quasi-Banach space can lose quasi-greediness upon passing to its Banach envelope, illustrating that envelope regularization can drastically alter greedy performance. Second, it exhibits an almost greedy Markushevich basis in a nonlocally convex quasi-Banach space that cannot be rearranged into a Schauder basis, underscoring the independence of greedy behavior from Schauder structure. Together, these results reveal that local convexity and envelope constructions critically shape greedy phenomena beyond classical Banach-space theory and highlight structural distinctions between Banach and quasi-Banach contexts.

Abstract

We construct two counterexamples that resolve long-standing open problems on greedy approximation theory with respect to bases, posed in [F. Albiac et al., Dissertationes Math. 560 (2021)] and restated in [F. Albiac, J. L. Ansorena, V. Temlyakov, J. Approx. Theory 307 (2025)]. Our first result exhibits a quasi-Banach space with an almost greedy basis which, when transported to the Banach envelope of , ceases to be quasi-greedy. This shows that the passage to the Banach envelope, although it preserves linear and lattice structure, may radically disrupt the performance of the thresholding greedy algorithm, to the extent that in some respects it could perform better in a quasi-Banach space than in its Banach envelope. Our second result constructs an almost greedy Markushevich basis in a nonlocally convex quasi-Banach space which fails to be a Schauder basis under any reordering. Together, these examples highlight that local convexity and the Banach envelope construction play an unexpectedly active role in shaping greedy approximation phenomena, revealing structural differences between Banach and quasi-Banach spaces that go beyond the classical theory of bases.
Paper Structure (4 sections, 40 theorems, 164 equations)

This paper contains 4 sections, 40 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Terminology and background
  3. Quasi-greediness does not pass to envelopes
  4. An almost greedy basis that fails to be a Schauder basis under any ordering

Key Result

Theorem 2.1

Let $({\mathbb{X}},\left\lVert\cdot\right\rVert)$ be a quasi-Banach space with modulus of concavity at most $\kappa$, and define $p\in(0,1]$ by $\kappa=2^{1/p-1}$. Then it:LpC:b holds with $C=4^{1/p}$.

Theorems & Definitions (75)

  • Theorem 2.1: see KPR1984
  • Lemma 2.2
  • proof
  • Theorem 2.3: cf. AABW2021*Theorem 4.1
  • Theorem 2.4: AABCO2024*Corollary 3.8
  • Theorem 2.5: AABW2021*Theorem 6.3
  • Lemma 2.6
  • proof
  • Theorem 2.7: see AABW2021*Section 9
  • Lemma 2.8
  • ...and 65 more