Non-linear approximation by $1$-greedy bases

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 $\Vert f-{\mathcal{G}}_m(f)\Vert \le C{\sigma }_m(f)$ 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, ${\sigma }_m(f)={\inf }_{y\in \mathbb{X}:|\text{supp}(y)|\le m}\Vert f-y\Vert$. 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 $f\in \mathbb{X}$.