We study high-dimensional nonlinear approximation of functions in H\"older-Nikol'skii spaces $H^\alpha_\infty(\mathbb{I}^d)$ on the unit cube $\mathbb{I}^d:=[0,1]^d$ having mixed smoothness, by parametric manifolds. The approximation error is measured in the $L_\infty$-norm. In this context, we explicitly constructed methods of nonlinear approximation, and give dimension-dependent estimates of the approximation error explicitly in dimension $d$ and number $N$ measuring computation complexity of the parametric manifold of approximants. For $d=2$, we derived a novel right asymptotic order of noncontinuous manifold $N$-widths of the unit ball of $H^\alpha_\infty(\mathbb{I}^2)$ in the space $L_\infty(\mathbb{I}^2)$. In constructing approximation methods, the function decomposition by the tensor product Faber series and special representations of its truncations on sparse grids play a central role.

翻译：我们研究H\"older-Nikol'skii 空间的高度非线性非线性近似值。 在单位立方体$\mathbb{I ⁇ d:[0,1 ⁇ d]美元,具有混合光滑度,用参数元体来测量。在此处,我们明确构建了非线性近似方法,并对近似误差作出视维度的估算,明确以美元和美元计值来计算相近方方数的计算复杂性。对于美元=2美元,我们得出了单位球的不连续元体的新型右线性亚值,即$Halpha ⁇ infty*infty(\mathbb{I ⁇ 2)美元。在构建近似度方法时,由索尔产品Faber系列的功能去complace 和其中层显示的阵列位置。