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Summability Methods for the Greedy Algorithm in Banach spaces

Miguel Berasategui, Pablo M. Berná, Stephen J. Dilworth, Denka Kutzarova

TL;DR

This paper studies the Thresholding Greedy Algorithm in Banach spaces and introduces two summability-based greedy-like bases: Cesàro quasi-greedy ($CQG$) and de la Vallée-Poussin quasi-greedy ($VPQG$), analyzed via greedy sums $G_n^{ ext{O}}(x)$ and their Cesàro/VPQG variants. It provides characterizations of $CQG$ and $VPQG$ bases in terms of convergence and pointwise boundedness, and develops auxiliary tools, including a bounding function $\Psi$ for controlling sums in subsequences. The work shows $VPQG$ bases are nearly unconditional and relates $CQG/VPQG$ to democracy and almost greediness, deriving conditions under which $VPQG$ implies quasi-greediness. A key consequence is a categorical proof that the uniform boundedness of greedy sums is equivalent to the convergence of the TGA, answering Wojtaszczyk's question, thereby extending the classical quasi-greedy framework. Together, these results provide new methods for nonlinear approximation in Banach spaces and deepen the connections between summability, democracy, and greedy-type bases.

Abstract

For the past 25 years, one of the most studied algorithms in the field of Nonlinear Approximation Theory has been the Thresholding Greedy Algorithm. In this paper, we propose new summability methods for this algorithm, generating two new types of greedy-like bases - namely Cesàro quasi-greedy and de la Vallée-Poussin-quasi-greedy bases. We analyze the connection between these types of bases and the well-known quasi-greedy bases, and leave some open problems for future research. In addition, as a consequence of our techniques for handling these summability methods, we answer a question posed by P. Wojtaszczyk in [16], by giving a categorial proof of equivalence between the uniform boundedness of the greedy sums and the convergence of the thresholding greedy algorithm.

Summability Methods for the Greedy Algorithm in Banach spaces

TL;DR

This paper studies the Thresholding Greedy Algorithm in Banach spaces and introduces two summability-based greedy-like bases: Cesàro quasi-greedy () and de la Vallée-Poussin quasi-greedy (), analyzed via greedy sums and their Cesàro/VPQG variants. It provides characterizations of and bases in terms of convergence and pointwise boundedness, and develops auxiliary tools, including a bounding function for controlling sums in subsequences. The work shows bases are nearly unconditional and relates to democracy and almost greediness, deriving conditions under which implies quasi-greediness. A key consequence is a categorical proof that the uniform boundedness of greedy sums is equivalent to the convergence of the TGA, answering Wojtaszczyk's question, thereby extending the classical quasi-greedy framework. Together, these results provide new methods for nonlinear approximation in Banach spaces and deepen the connections between summability, democracy, and greedy-type bases.

Abstract

For the past 25 years, one of the most studied algorithms in the field of Nonlinear Approximation Theory has been the Thresholding Greedy Algorithm. In this paper, we propose new summability methods for this algorithm, generating two new types of greedy-like bases - namely Cesàro quasi-greedy and de la Vallée-Poussin-quasi-greedy bases. We analyze the connection between these types of bases and the well-known quasi-greedy bases, and leave some open problems for future research. In addition, as a consequence of our techniques for handling these summability methods, we answer a question posed by P. Wojtaszczyk in [16], by giving a categorial proof of equivalence between the uniform boundedness of the greedy sums and the convergence of the thresholding greedy algorithm.

Paper Structure

This paper contains 11 sections, 17 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Main definitions and Notation
  3. Sets
  4. Unconditionality
  5. The TGA and Cesàro-summability
  6. Indicator sums
  7. Caracterization of CQG and VPQG bases
  8. Some technical results
  9. VPQG bases and convergence through subsequences
  10. Cesàro quasi-greedy bases, de la Valée Poussin quasi-greedy bases, and democracy.
  11. Open questions

Key Result

Lemma 3.1

Let $\mathcal{X}$ be a basis for a Banach space $\mathbb{X}$, let $A\subset \mathbb{N}$ be a nonempty set, $K_1$ a positive constant, and $\left(t_n\right)_{n\in \mathbb{N}_0}$ a sequence of positive numbers. For each $m\in A$, let $\left(b_{m,j}\right)_{1\le j\le m}$ be scalars such that Suppose that for each $x\in \mathbb{X}$ there is $c_x>0$ such that, for each $m\in A$ there is a greedy order

Theorems & Definitions (50)

  • Definition 2.1
  • Definition 2.2: KT1999
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 40 more