论文标题
$ l^p（\ log l）^α$ space的弱chebyshev贪婪算法（WCGA）
The Weak Chebyshev Greedy Algorithm (WCGA) in $L^p (\log L)^α$ spaces
论文作者
论文摘要
我们提出了一些有关Lebesgue型不平等的新结果，这些不平等现象是均匀平滑的Banach空间$ \ Mathbb {x} $的弱chebyshev贪婪算法（WCGA）。首先，我们将Temlyakov的结果推广到涵盖平滑度和所谓A3参数不一定是功率功能的情况。其次，我们将此新定理应用于Zygmund空间$ \ mathbb {x} = l^p（\ log l）^α$，$ 1 <p <p <\ infty $和$α\ in \ mathbb {r} $ in \ mathbb {r} $，并在使用$ n $ n $ n $ n $ -n $ n $ -n $ -n $ n $ n $ -n $ -n $ n $ n $ n $ -n $ -n $ ϕ（n）= o（n^{\ max \ {1,2/p'\}}} \，（\ log n）^{|α| p'}）$。此外，当$ p \ leq 2 $时，此数量很清晰。最后，还给出了三角系统的$ ϕ（n）$的表达式，在$ l^2（\ log l）^α$的特殊情况下，带有$α> 0 $的表达式为$ $ ϕ（n）\ oft \ log log（\ log n）$。
We present some new results concerning Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm (WCGA) in uniformly smooth Banach spaces $\mathbb{X}$. First, we generalize a result of Temlyakov to cover situations in which the modulus of smoothness and the so called A3 parameter are not necessarily power functions. Secondly, we apply this new theorem to the Zygmund spaces $\mathbb{X}=L^p(\log L)^α$, with $1<p<\infty$ and $α\in\mathbb{R}$, and show that, when the Haar system is used, then optimal recovery of $N$-sparse signals occurs when the number of iterations is $ϕ(N)=O(N^{\max\{1,2/p'\}} \,(\log N)^{|α| p'})$. Moreover, this quantity is sharp when $p\leq 2$. Finally, an expression for $ϕ(N)$ in the case of the trigonometric system is also given, which in the special case of $L^2(\log L)^α$, with $α>0$, takes the form $ϕ(N)\approx \log(\log N)$.
