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 Hölder-Nikol'skii spaces H α ∞ (I d ) on the unit cube I d := [0, 1] d having mixed smoothness, by parametric manifolds. The approximation error is measured in the L ∞ -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 ∞ (I 2 ) in the space L ∞ (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.