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.

翻译：在本文中,我们展示了统一规范中多变量函数类别无症状特征的结果。 我们的主要兴趣在于将功能与不大于1/2美元的混合光滑参数相近。 我们的焦点将放在最优秀的美元中期三角近似以及统一规范中科尔莫戈洛夫和英特罗比数字的衰变上。 事实证明, 这些数量共享一些基本抽象属性, 比如在真实的内推下的行为, 从而可以同时处理它们。 我们首先证明对有限级的级调控操作员的估计值, 其范围为一步双曲十字。 这些结果意味着一个众所周知的分解技术嵌入相应功能空间的界限。 科尔莫戈洛夫数字的衰减直接影响到在文献中最近的结果不适用的情况下以$_2美元进行抽样回收的问题。 因为相应的近似数字无法对等相加。