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.
This paper determines the exact error order on optimization of adaptive direct methods of approximate solution of the class of Fredholm integral equations of the second kind with kernel belonging to the anisotropic Sobolev classes, and also gives an optimal algorithm.
The order of computational complexity of all bounded linear functional ap proximation problem is determined for the generalized Sobolev class Wp?（Id）, Nikolskii class H|∞k（Id） in the worst （deterministic）, stochastic and average case setting, from which it is concluded that the bounded linear functional approximation problem for the classes Wp?（Id） and H∞k（Id） is intractable in worst case setting, but is tractable with respect to stochastic and average case setting.
In this paper, we introduce a problem of the optimization of approximate solutions of operator equations in the probabilistic case setting, and prove a general result which connects the relation between the optimal approximation order of operator equations with the asymptotic order of the probabilistic width. Moreover, using this result, we determine the exact orders on the optimal approximate solutions of multivariate Freldholm integral equations of the second kind with the kernels belonging to the multivariate Sobolev class with the mixed derivative in the probabilistic case setting.
The truncation error associated with a given sampling representation is defined as the difference between the signal and on approximating sumutilizing a finite number of terms. In this paper we give uniform bound for truncation error of bandlimited functions in the n dimensional Lebesgue space Lp(Rn) associated with multidimensional Shannon sampling representation.
