This paper determines the exact error order on optimization of adaptive directmethods of approximate solution of the class of Fredholm integral equations of the secondkind with kernel belonging to the anisotropic Sobolev classes, and also gives an optimalalgorithm.
The order of computational complexity of all bounded linear functional approximation problem is determined for the generalized Sobolev class Wp∧(Id), Nikolskii class Hk∞(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 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 Preldholm 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 an 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(R^n) associated with multidimensional Shannon sampling representation.
