论文标题
在统一规范中具有较小混合平滑度的功能的近似
Approximation of functions with small mixed smoothness in the uniform norm
论文作者
论文摘要
在本文中，我们介绍了统一规范中多元函数类别的渐近特征。我们的主要兴趣是混合光滑参数不大于$ 1/2 $的功能的近似。我们的重点将放在最佳的$ M $ $ term Trigonometric近似以及Kolmogorov的衰减以及均匀规范中的熵数的行为。事实证明，这些数量具有一些基本的抽象属性，例如它们在实际插值下的行为，以便可以同时对待它们。我们首先要证明对有限排名卷积运算符的估计，该运算符在阶跃双曲线交叉中范围。这些结果暗示着通过众所周知的分解技术嵌入相应功能空间的边界。 Kolmogorov数字的衰减对在文献最新结果不适用的情况下对$ l_2 $采样恢复的问题有直接影响，因为相应的近似编号不是可以总结的。
In this paper we present results on asymptotic characteristics of multivariate function classes in the uniform norm. Our main interest is the approximation of functions with mixed smoothness parameter not larger than $1/2$. Our focus will be on the behavior of the best $m$-term trigonometric approximation as well as the decay of Kolmogorov and entropy numbers in the uniform norm. It turns out that these quantities share a few fundamental abstract properties like their behavior under real interpolation, such that they can be treated simultaneously. We start with proving estimates on finite rank convolution operators with range in a step hyperbolic cross. These results imply bounds for the corresponding function space embeddings by a well-known decomposition technique. The decay of Kolmogorov numbers have direct implications for the problem of sampling recovery in $L_2$ in situations where recent results in the literature are not applicable since the corresponding approximation numbers are not square summable.
