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.     

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

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

The Average Widths and Non-linear Widths of the Classes of Multivariate Functions with Bounded Moduli of Smoothness

The classes of the multivariate functions with bounded moduli on Rd and Td are given and their average σ-widths and non-linear n-widths are discussed. The weak asymptotic behaviors are established for the corresponding quantities.     

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.     

μ-AVERAGE N-WIDTHS ON THE WIENER SPACE

In this paper we investigate the asymptotic bebaviour of μ-average n-widths of integral operator Kon the Wiener space. where K is the inverse operator of an ordinary linear differential operator L of orderm. For 1≤p.q<∞ C_n~a(K,W)_(p,q)a_n~a(K,W)_(p,q)n~(-(m)-(1/2)and for p∈(1,∞), q∈(2,∞) d_n~a(K; W)_(p.q)n~(-(m)-(1/2)).     

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≤∞.     

Bernstein <Emphasis Type="Italic">n</Emphasis>-widths of Besov embeddings on Lipschitz domains

In this paper, using an equivalent characterization of the Besov space by its wavelet coefficients and the discretization technique due to Maiorov, we determine the asymptotic degree of the Bernstein n-widths of the compact embeddings Bq0s＋t（Lp0（Ω））→Bq1s（Lp1（Ω））, t〉max{d（1/p0-1/p1）, 0}, 1 ≤ p0, p1, q0, q1 ≤∞,where Bq0s＋t（Lp0（Ω）） is a Besov space defined on the bounded Lipschitz domain Ω ？ Rd. The results we obtained here are just dual to the known results of Kolmogorov widths on the related classes of functions.     

ON n- WIDTHS OF PERIODIC FUNCTIONS

Let ~(r,1)_p={f:f~(r-1) is abs. cont. on I=[a, b], f is periodie with period H(=b-a),f(x_1)=0,‖f~((r))‖_p≤0}, where x_1 is any fixed point in [a, b]. The author finds the Kolmogorov, Gel'land, linear, and Berustein n-widths of ~(r,1)_p in L~p(I) for n odd, ∞>p>1. The optimal subspaces and operators are also found.     

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.     

Boundary Shape Control of the Navier-Stokes Equations and Applications

In this paper, the geometrical design for the blade's surface in an impeller or for the profile of an aircraft, is modeled from the mathematical point of view by a boundary shape control problem for the Navier-Stokes equations. The objective function is the sum of a global dissipative function and the power of the fluid. The control variables are the geometry of the boundary and the state equations are the Navier-Stokes equations. The Euler-Lagrange equations of the optimal control problem are derived, which are an elliptic boundary value system of fourth order, coupled with the Navier-Stokes equations. The authors also prove the existence of the solution of the optimal control problem, the existence of the solution of the Navier-Stokes equations with mixed boundary conditions, the weak continuity of the solution of the Navier-Stokes equations with respect to the geometry shape of the blade's surface and the existence of solutions of the equations for the Gateaux derivative of the solution of the Navier-Stokes equations with respect to the geometry of the boundary.     

Optimal nonlinear approximation

We introduce a definition of nonlinear n-widths and then determine the n-widths of the unit ball of the Sobolev spaceW _p ^r inL _q. We prove that in the sense of these widths the manifold of splines of fixed degree with n free knots is optimal for approximating functions in these Sobolev spaces.This author was supported by NSF Grant DMS 8620108This author was supported by NSF Grant DMS 8803585     

广义周期Besov类在周期Sobolev空间中的宽度

本文给出了一种广义周期Besov类在周期Sobolev空间中的n-宽度的弱渐近估计。     

Comparison Theorems of Kolmogorov Type for Classes Defined by Cyclic Variation Diminishing Operators and Their Application

Presenting a unified approach, we establish a Kolmogorov-type comparison theorem for classes of 2π-periodic functions defined by a special class of operators having certain oscillation properties, which include the classical Sobolev class of 2π-periodic functions, the Achieser class, and the Hardy-Sobolev class as examples. Then, using these results, we prove a Taikov-type inequality, and calculate the exact values of the Kolmogorov, Gel'fand, linear, and information n-widths of these classes of functions in the space L_q, which is the classical Lebesgue integral space of 2π-periodic functions with the usual norm.     

最坏框架与平均框架下区间[1,1]上带Jacobi权的函数逼近

本文研究最坏框架和平均框架下区间[1,1]上带Jocobi权(1 x)α(1+x)β,α,β1/2的函数逼近问题.在最坏框架下,本文得到加权Sobolev空间BWr p,α,β在Lq,α,β(1 q∞)空间尺度下的Kolmogorov n-宽度和线性n-宽度的渐近最优阶,其中Lq,α,β(1 q∞)表示区间[1,1]上带Jacobi权的加权Lq空间.在平均框架下,本文研究具有Gauss测度的加权Sobolev空间Wr2,α,β被多项式子空间和Fourier部分和算子在Lq,α,β(1 q∞)空间尺度下的最佳逼近问题,得到平均误差估计的渐近阶.我们发现,在平均框架下,多项式子空间和Fourier部分和算子在Lq,α,β(1 q2+22 max{α,β}+1)空间尺度下是渐近最优的线性子空间和渐近最优的线性算子.     

One-sided N-widths of some classes of periodic functions

In this paper,we determine the exact estimation of one-sided n-widthsd_n~+(w_q~r(p_r),L_1).q=1,∝,where W_q~r(P_r)is a class of differentiable 2π-periodic functions,and is determined by a differential opera-tor P_r(D)=D~σ∏_(j=t)~l(D-α_j),σ,j∈N,σ≥2,x_j≠0,x_j∈R.     

N-widths of Ω r ′ in L p

Let Ω _ϕ ^r ={f:f ^(r-1) abs. cont. on [0,1], ‖q_r(D)f‖_p≤1, f^(2K+σ) (0)=f^(2K+σ)=0, (k)=0,...,l-1}. where , and I is an identical operator. Denote Kolmogorov, linear, Geelfand and Bernstein n-widths of Ω _ϕ ^r in L^p byd _n (Ω _ϕ ^r ;L ^p ),δ _n (Ω _ϕ ^r ;L ^p ),d ⁿ (Ω _p ^r ;L ^p ) andb _n (Ω _p ^r ;L ^p ), respectively. In this paper, we find a method to get an exact estimation of these n-widths. Related optimal subspaces and an optimal linear operator are given. For another subset , similar results are also derrived.