混合光滑Hölder-Nikol'skii空间中参数流形的高维非线性逼近
High-dimensional nonlinear approximation by parametric manifolds in Hölder-Nikol'skii spaces of mixed smoothness
摘要：
研究了混合光滑的单位立方体$\mathbb{I}^d:=[0,1]^d$上Hölder-Nikol'skii空间$H^α_\infty(\mathbb{I}^d)$中函数的高维参数流形的非线性逼近.近似误差以$L_\infty$范数度量。在此背景下，我们显式地构造了非线性逼近方法，并给出了逼近误差的维数依赖估计，其中维数为$d$，维数为$N$，参数流形的计算复杂性度量.对于$d=2$，我们在空间$L_\infty(\mathbb{I}^2)$中导出了$H^α_\infty(\mathbb{I}^2)$的单位球的非连续流形$N$ -宽度的一个新的右渐近阶.在构造逼近方法时，函数的张量积Faber级数分解及其在稀疏网格上截断的特殊表示起着核心作用。
英文摘要：
We study high-dimensional nonlinear approximation of functions in Hölder-Nikol'skii spaces $H^α_\infty(\mathbb{I}^d)$ 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 $H^α_\infty(\mathbb{I}^2)$ 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.
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