THE N-WIDTHS FOR A GENERALIZED PERIODIC BESOV CLASSES

In this paper, an extension of Besov classes of periodic functions on Td is given. The weak asymptotic results concerning the Kolmogorov n-widths, the linear n-widths, and the Gel＇fand n-widths are obtained, respectively.

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.

THE GEL'FAND n-WIDTHS OF MULTIVARIATE PERIODIC BESOV CLASSES

The Gel'fand n-widths of Besov classes are considered. The weak asymptotic behavior is established for the corresponding quantity.

THI GENERALIZED RAYLEIGH SPECTRA AND KOLMOGOROV n-WIDTHS

Let F(x):R^m→R^m be an odd,continuously differentiable homogeneous map.The pa- per is devoted to the critical points of the generalized Rayleigh ratio||F(x)||_(l_q^m)/||x||_(l_p^m)and connected with some problems of the approximation theory.We find the lower bound for Kolmogorov n-width d_n(F(Bl_p^m),l_q^m).

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

In this paper we will show that if an approximation process {Ln}n∈N is shape- preserving relative to the cone of all k-times differentiable functions with non-negative k-th derivative on [0,1], and the operators Ln are assumed to be of finite rank n, then the order of convergence of D^kLnf to D^kf 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 LP[0, 1], p ∈ N.

Bernstein n-Widths for Classes of Convolution Functions with Kernels Satisfying Certain Oscillation Properties

In this paper, we consider some classes of 2π-periodic convolution functions Bp, and Kp with kernels having certain oscillation properties, which include the classical Sobolev class as special case. With the help of the spectral of nonlinear integral equations, we determine the exact values of Bernstein n-width of the classes Bp, Kp in the space Lp for 1 〈 p 〈 ∞.

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） = DσПlj=1 （D2 - tj2I） 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≤∞.

On the 2n-Widths of a Periodic Sobolev Class

1 Introduction Let Q（x）be a polynomial with real coefficients and the Sobolev class Wp（Q（D））be the set of continuous 2π-periodic functions f（x） for which f（degQ-1） is absolutely continuous and ‖Q（D）f‖p≤1, where degQ is the degree of Q, D=d/dt, and ‖·‖p is the usual Lp[0, 2π]-

n-WIDTHS OF SOME CLASSES OF PERIODIC FUNCTIONS

Let p = {a +G*h|a∈R, h⊥1, ||h||p≤1}, where G is a B-kernel. We obtain the exactvalues of the d2n（Bp, Lp）, d2n（Bp,Lp） and δ2n（Bp,Lp） for p∈（1,+∞）/{2}. Furthermore. weidentify some optimal subspaces for d2n and d2n respectively, and construct an optimallinear operator of rank 2n for δ2n, from which we answer affirmatively Pinkus Conjecture,i.e. ?p≥q≥1, W2n={a +sum from i=0 to （2n-1） aiG（x-iπ/n）a,ai∈R,sum from i=0 to（2n-1） ai=0} is an optimal subspace ford2n（?p,Lp） for p=q.

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.

AVERAGE n-WIDTH OF POINT SET IN HILBERT SPACE

Given a separable real Banach space F and a separable real Hilbert space X, we denote by S a bounded linear operator from F into X. Let μ be a Gaussian measure defined on the Borel field of F with mean element m_μ= 0 and correlation operator C_μ. By [1], C_μ is a bounded linear operator from F~*→F such

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.