论文标题
非线性近似值$ 1 $ - 绿色基础
Non-linear approximation by $1$-greedy bases
论文作者
论文摘要
贪婪的基础理论始于1999年，当时S. V. Konyagin和V. N. Temlyakov在\ cite {kt}中引入了著名的阈值贪婪算法。自今年以来，文献中出现了不同的类似贪婪的基础，例如：准怪兽，几乎是怪兽和贪婪的基础。本文的目的是介绍1个蛋白质基础的一些新特征。 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 $\Vert f-\mathcal G_m(f)\Vert\leq Cσ_m(f)$ for每个$ f \ in \ mathbb x $和\ mathbb n $中的每个$ m \，其中$σ_m（f）$是$ f $的近似值中最好的$ m $ th错误，即$ f $，$σ_m（f）= \ inf_ in \ in \ in \ in \ in \ nathbb {x} m} \ vert f-y \ vert $。在这里，当$ c = 1 $显示$ \ vert f- \ mathcal g_1（f）\ vert =σ_1（f）$时，我们将注意力集中在$ f \ in \ mathbb x $时。
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 $\Vert f-\mathcal G_m(f)\Vert\leq Cσ_m(f)$ for every $f\in\mathbb X$ and every $m\in\mathbb N$, where $σ_m(f)$ is the best $m$th error in the approximation for $f$, that is, $σ_m(f)=\inf_{y\in\mathbb{X} : \vert \text{supp}(y)\vert\leq m}\Vert f-y\Vert$. Here, we focus our attention when $C=1$ showing that a basis is 1-greedy if and only if $\Vert f-\mathcal G_1(f)\Vert=σ_1(f)$ for every $f\in\mathbb X$.
