High-dimensional nonlinear approximation by parametric manifolds in H\'older-Nikol'skii spaces of mixed smoothness
| Autor: | Dũng, Dinh, Nguyen, Van Kien |
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| Rok vydání: | 2021 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 25 pages |
| Databáze: | arXiv |
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