High-dimensional nonlinear approximation by parametric manifolds in Hölder-Nikol'skii spaces of mixed smoothness

Abstract: We study high-dimensional nonlinear approximation of functions in H\"older-Nikol'skii spaces $H<sup>\alpha_\infty(\mathbb{I}<sup>d)$ on the unit cube $\mathbb{I}<sup>d:=[0,1]<sup>d$ 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 $H<sup>\alpha_\infty(\mathbb{I}<sup>2)$ in the space $L_\infty(\mathbb{I}<sup>2)$. 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.