In this paper, the non-linear approximation on the class of multivariate functions with bounded mixed derivatives is investigated, and the asymptotic degree of the non-linear width on this class is determined.
Temlyakov considered the optimal recovery on the classes of functions with bounded mixed derivative in the Lp metrics and gave the upper estimates of the optimal recovery errors. In this paper, we determine the asymptotic orders of the optimal recovery in Sobolev spaces by standard information, i.e., function values, and give the nearly optimal algorithms which attain the asymptotic orders of the optimal recovery.
In this paper, we study the complexity of information of approximation problem on the multivariate Sobolev space with bounded mixed derivative MW p,α r ( $ \mathbb{T}^d $ ), 1 < p < ∞, in the norm of L q ( $ \mathbb{T}^d $ ), 1 < q < ∞, by adaptive Monte Carlo methods. Applying the discretization technique and some properties of pseudo-s-scale, we determine the exact asymptotic orders of this problem.
