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Approximation spaces, greedy classes and Lorentz spaces

Miguel Berasategui, Pablo M. Berná, Andrea García

TL;DR

This work characterizes non-linear approximation spaces for a broad class of bases, including almost greedy ones, in terms of weighted Lorentz spaces and identifies when greedy and classical approximation spaces coincide. By developing embeddings between weighted Lorentz sequence spaces and greedy/approximation spaces under weaker hypotheses (notably truncation quasi-greediness and the APP property), the authors extend prior results beyond unconditional bases and standard weights. They establish necessary and sufficient conditions linking democracy and superdemocracy to the equality of greedy and approximation spaces, using Bernstein/Jackson-type inequalities and the finite dimensional separation property. The results generalize classical $k^{\alpha}$-weighted characterizations to wider weight classes and provide a unified framework for understanding how greedy-type methods approximate elements in quasi-Banach spaces. The analysis has implications for applying Lorentz-space techniques to non-linear approximation problems in harmonic analysis and related areas.

Abstract

We characterize the approximation spaces of a broad class of bases - which includes almost greedy bases - in terms of weighted Lorentz spaces. For those bases, we also find necessary and sufficient conditions under which the approximation spaces and greedy classes are the same.

Approximation spaces, greedy classes and Lorentz spaces

TL;DR

This work characterizes non-linear approximation spaces for a broad class of bases, including almost greedy ones, in terms of weighted Lorentz spaces and identifies when greedy and classical approximation spaces coincide. By developing embeddings between weighted Lorentz sequence spaces and greedy/approximation spaces under weaker hypotheses (notably truncation quasi-greediness and the APP property), the authors extend prior results beyond unconditional bases and standard weights. They establish necessary and sufficient conditions linking democracy and superdemocracy to the equality of greedy and approximation spaces, using Bernstein/Jackson-type inequalities and the finite dimensional separation property. The results generalize classical -weighted characterizations to wider weight classes and provide a unified framework for understanding how greedy-type methods approximate elements in quasi-Banach spaces. The analysis has implications for applying Lorentz-space techniques to non-linear approximation problems in harmonic analysis and related areas.

Abstract

We characterize the approximation spaces of a broad class of bases - which includes almost greedy bases - in terms of weighted Lorentz spaces. For those bases, we also find necessary and sufficient conditions under which the approximation spaces and greedy classes are the same.

Paper Structure

This paper contains 9 sections, 14 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. General setting and notation
  3. Sets.
  4. Indicator sums and democracy-like properties.
  5. Thresholding Greedy Algorithm.
  6. Weights.
  7. Greedy and approximation classes.
  8. Approximation classes and Lorentz spaces
  9. Classical, greedy and Chebyshev-greedy approximation classes.

Key Result

Theorem 2.8

Let $\mathbb{X}$ be a quasi-Banach space and $\mathcal{B}$ a Markushevich basis of $\mathbb{X}$. The following are equivalent:

Theorems & Definitions (39)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Definition 2.10
  • ...and 29 more