N-Widths and Average Widths of Besov Classes in Sobolev Spaces

In this paper, we consider the n-widths and average widths of Besov classes in the usual Sobolev spaces. The weak asymptotic results concerning the Kolmogorov n-widths, the linear n-widths, the Gel＇fand n-widths, in the Sobolev spaces on T^d, and the infinite-dimensional widths and the average widths in the Sobolev spaces on Ra are obtained, respectively.     

Fréchet spaces with unconditional base

A nonnormable ?-space possessing an unconditional basis is not necessarily isomorphic to a generalized Köthe space.     

Average optimization of the approximate solution of operator equations and its application

In this paper, a definition of the optimization of operator equations in the average case setting is given. And the general result (Theorem 1) about the relevant optimization problem is obtained. This result is applied to the optimization of approximate solution of some classes of integral equations     

On n-widths of a Sobolev function class in Orlicz spaces

This paper considers the problem of n-widths of a Sobolev function class Ωr∞ determined by Pr(D) = Dσ∏lj =1(D2- t2jI) in Orlicz spaces. The relationship between the extreme value problem and width theory is revealed by using the methods of functional analysis. Particularly, as σ = 0 or σ = 1, the exact values of Kolmogorov's widths, Gelfand's widths, and linear widths are obtained respectively, and the related extremal subspaces and optimal linear operators are given.     

Relative n-widths of periodic convolution classes with NCVD-kernel and B-kernel

In this paper,we consider the relative n-widths of two kinds of periodic convolution classes,Kp(K) and Bp(G),whose convolution kernels are NCVD-kernel K and B-kernel G. The asymptotic estimations of Kn(Kp(K),Kp(K))q and Kn(Bp(G),Bp(G))q are obtained for p=1 and ∞,1≤ q≤∞.     

OPTIMAL ALGORITHMS FOR DIAGONAL OPERATORS ON N-WIDTHS IN DIFFERENT COMPUTATIONAL SETTING -In Memory of Professor Sun Yongsheng

In this paper, we give some optimal algorithms for diagonal operator T from space lp（1≤p≤2） to l2 on n-widths in different computational setting.     

ESTIMATES OF LINEAR RELATIVE n-WIDTHS IN L~p [0,1]

In this paper we will show that if an approximation process { L n }n ∈ N is shapepreserving relative to the cone of all k-times differentiable functions with non-negative k-th derivative on [0,1],and the operators L n are assumed to be of finite rank n,then the order of convergence of D k L n f to D k f cannot be better than n 2 even for the functions x k,x k+1,x k+2 on any subset of [0,1] with positive measure.Taking into account this fact,we will be able to find some asymptotic estimates of linear relative n-width of sets of differentiable functions in the space L p [0,1],p ∈ N.