论文标题
稀疏集合的贪婪近似算法
Greedy approximation algorithms for sparse collections
论文作者
论文摘要
我们描述了一种贪婪的算法，该算法近似于一系列一般集合的Carleson常数。在一般环境中，近似值具有对数损失，但最佳达到恒定，只有轻度的几何假设。该算法的建设性提供了有关稀疏集合几乎分散结构的其他信息。 作为应用程序，我们在每个维度中的轴平行矩形的收集给出了三个结果。首先是Hänninen首先显示的Carleson和稀疏集合之间等效的建设性证明。第二个是一个结构定理，证明每个集合$ \ MATHCAL {e} $可以分为$ \ MATHCAL {O}（n）$稀疏的亚家族，其中$ n $是$ \ MATHCAL {E} $的Carleson常数。我们还举例说明，当几何假设被删除时，这种分解是不可能的。第三个应用程序是Carleson常数的表征，仅涉及$ l^{1，\ infty} $估计。
We describe a greedy algorithm that approximates the Carleson constant of a collection of general sets. The approximation has a logarithmic loss in a general setting, but is optimal up to a constant with only mild geometric assumptions. The constructive nature of the algorithm gives additional information about the almost-disjoint structure of sparse collections. As applications, we give three results for collections of axis-parallel rectangles in every dimension. The first is a constructive proof of the equivalence between Carleson and sparse collections, first shown by Hänninen. The second is a structure theorem proving that every collection $\mathcal{E}$ can be partitioned into $\mathcal{O}(N)$ sparse subfamilies where $N$ is the Carleson constant of $\mathcal{E}$. We also give examples showing that such a decomposition is impossible when the geometric assumptions are dropped. The third application is a characterization of the Carleson constant involving only $L^{1,\infty}$ estimates.
