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On sequential greedy-type bases

Miguel Berasategui, Pablo M. Berná, Hung Viet Chu

TL;DR

The paper studies sequential greedy-type bases by introducing the $\mathcal{F}_{(a_n)}$-almost greedy and related strong/minimum partially greedy notions, where projection targets are chosen from interval-like sets determined by a fixed gap sequence $(a_n)$. It proves that $\mathcal{F}_{(a_n)}$-almost greediness coincides with classical almost greediness if and only if $(a_n)$ is bounded, and it establishes analogous results for strong partially greedy bases. When $(a_n)$ is unbounded, the authors construct bases that are $\mathcal{F}_{(a_n)}$-almost greedy but not almost greedy, showing that the new framework can yield strict separations; they also develop a hierarchical landscape (including $\mathcal{F}_{(a_n)}$-minimum partially greedy) that can interpolate between almost greedy and strong partially greedy under various conditions. The work relies on a structural framework using $N$-covering and $N$-sliding properties to connect family-dependent greedy notions to classical ones, providing insight into how gap patterns influence greedy performance in $p$-Banach spaces and Schauder bases.

Abstract

It is known that a basis is almost greedy if and only if the thresholding greedy algorithm gives essentially the smallest error term compared to errors from projections onto intervals or in other words, consecutive terms of $\mathbb{N}$. In this paper, we fix a sequence $(a_n)_{n=1}^\infty$ and compare the TGA against projections onto consecutive terms of the sequence and its shifts. We call the corresponding greedy-type condition the $\mathcal{F}_{(a_n)}$-almost greedy property. Our first result shows that the $\mathcal{F}_{(a_n)}$-almost greedy property is equivalent to the classical almost greedy property if and only if $(a_n)_{n=1}^\infty$ is bounded. Then we establish an analog of the result for the strong partially greedy property. Finally, we show that under a certain projection rule and conditions on the sequence $(a_n)_{n=1}^\infty$, we obtain a greedy-type condition that lies strictly between the almost greedy and strong partially greedy properties.

On sequential greedy-type bases

TL;DR

The paper studies sequential greedy-type bases by introducing the -almost greedy and related strong/minimum partially greedy notions, where projection targets are chosen from interval-like sets determined by a fixed gap sequence . It proves that -almost greediness coincides with classical almost greediness if and only if is bounded, and it establishes analogous results for strong partially greedy bases. When is unbounded, the authors construct bases that are -almost greedy but not almost greedy, showing that the new framework can yield strict separations; they also develop a hierarchical landscape (including -minimum partially greedy) that can interpolate between almost greedy and strong partially greedy under various conditions. The work relies on a structural framework using -covering and -sliding properties to connect family-dependent greedy notions to classical ones, providing insight into how gap patterns influence greedy performance in -Banach spaces and Schauder bases.

Abstract

It is known that a basis is almost greedy if and only if the thresholding greedy algorithm gives essentially the smallest error term compared to errors from projections onto intervals or in other words, consecutive terms of . In this paper, we fix a sequence and compare the TGA against projections onto consecutive terms of the sequence and its shifts. We call the corresponding greedy-type condition the -almost greedy property. Our first result shows that the -almost greedy property is equivalent to the classical almost greedy property if and only if is bounded. Then we establish an analog of the result for the strong partially greedy property. Finally, we show that under a certain projection rule and conditions on the sequence , we obtain a greedy-type condition that lies strictly between the almost greedy and strong partially greedy properties.
Paper Structure (7 sections, 27 theorems, 168 equations)

This paper contains 7 sections, 27 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Framework for an infinite family $\mathcal{F}$ of finite subsets
  4. Equivalence requires bounded gaps
  5. On $\mathcal{F}_{(a_n)}$-strong partially greedy bases
  6. Between the almost greedy and strong partially greedy properties
  7. Appendix

Key Result

Theorem 2.3

Fix an $N$-covering and $M$-sliding family $\mathcal{F}$. Then a basis $\mathcal{B}$ is $\mathcal{F}$-almost greedy if and only if it is almost greedy.

Theorems & Definitions (71)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Definition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • ...and 61 more