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