Non-linear approximation by $1$-greedy bases

TL;DR

This paper characterizes 1-greedy bases in Banach spaces, showing they are exactly those for which the greedy approximation with one element achieves the best possible error, refining understanding of greedy algorithms.

Contribution

It introduces new characterizations of 1-greedy bases, establishing an equivalence between 1-greediness and optimal one-element approximation in Banach spaces.

Findings

1-greedy bases are characterized by optimal one-element approximation.

The paper provides new criteria for identifying 1-greedy bases.

It refines the theoretical understanding of greedy algorithms in Banach spaces.

Abstract

The theory of greedy-like bases started in 1999 when S. V. Konyagin and V. N. Temlyakov introduced in \cite{KT} the famous Thresholding Greedy Algorithm. Since this year, different greedy-like bases appeared in the literature, as for instance: quasi-greedy, almost-greedy and greedy bases. The purpose of this paper is to introduce some new characterizations of 1-greedy bases. Concretely, given a basis $\mathcal B=(\mathbf x_n)_{n\in\mathbb N}$ in a Banach space $\mathbb X$, we know that $\mathcal B$ is $C$-greedy with $C>0$ if $\Vert f-\mathcal G_m(f)\Vert\leq C\sigma_m(f)$ for every $f\in\mathbb X$ and every $m\in\mathbb N$, where $\sigma_m(f)$ is the best $m$th error in the approximation for $f$, that is, $\sigma_m(f)=\inf_{y\in\mathbb{X} : \vert \text{supp}(y)\vert\leq m}\Vert f-y\Vert$. Here, we focus our attention when $C=1$ showing that a basis is 1-greedy if and only if $\Vert f-\mathcal G_1(f)\Vert=\sigma_1(f)$ for every $f\in\mathbb X$.