Average Width and Optimal Recovery of Some Anisotropic Classes of Smooth Functions Defined on the Euclidean Space R~d

§ 1.Introduction and Main Results　 In[1 ] ,[2 ] ,the authors studied some problems of optimal recovery of functions de-fined on a cube,for a class of functions with partial derivatives of a fixed order havingmoduli of continuity not exceeding a given modules of continuity,and for the unit ballsSHαp in the spaces Hαp satisfying the mixed Holder conditionα,respectively.They ob-tained some weak asymptotic results.　 In[3 ] ,[4 ] and[5] ,Magarill-Il' yaev,Liu and Sun studied some proble…     

Some extremal properties of multivariate polynomial splines in the metric Lp (Rd )

We constructed a kind of continuous multivariate spline operators as the approximation tools of the multivariate functions on the Bd instead of the usual multivariate cardinal interpolation oper-ators of splines, and obtained the approximation error by this kind of spline operators. Meantime, by the results, we also obtained that the spaces of multivariate polynomial splines are weakly asymptoti-cally optimal for the Kolmogorov widths and the linear widths of some anisotropic Sobolev classes of smooth functions on Bd in the metric Lp(Bd).     

Some extremal properties of multivariate polynomial splines in the metric Lp(ℝd)

We constructed a kind of continuous multivariate spline operators as the approximation tools of the multivariate functions on the (?^d instead of the usual multivariate cardinal interpolation operators of splines, and obtained the approximation error by this kind of spline operators. Meantime, by the results, we also obtained that the spaces of multivariate polynomial splines are weakly asyrnptotically optimal for the Kolrnogorov widths and the linear widths of some anisotropic Sobolev classes of smooth functions on (?^d in the metric L_p((?^d).     

WIDTHS AND AVERAGE WIDTHS OF SOBOLEV CLASSES

This paper concerns the problem of the Kolmogorov n-width, the linear re-width, the Gel'fand n-width and the Bernstein re-width of Sobolev classes of the periodic multivariate functions in the space Lp(Td) and the average Bernstein o-width, average Kolmogorov o-widths, the average linear o-widths of Sobolev classes of the multivariate functions in the space LP(R ), where p = (p1,…,pd), 1 < Pj < ∞o, j = 1,2,…,d, or pj = ∞,j = 1,2,…, d. Their weak asymptotic behaviors are established for the corresponding quantities.     

Some extremal properties of multivariate polynomial splines in the metric Lp(?d)

We constructed a kind of continuous multivariate spline operators as the approximation tools of the multivariate functions on the (ℝ^d instead of the usual multivariate cardinal interpolation operators of splines, and obtained the approximation error by this kind of spline operators. Meantime, by the results, we also obtained that the spaces of multivariate polynomial splines are weakly asyrnptotically optimal for the Kolrnogorov widths and the linear widths of some anisotropic Sobolev classes of smooth functions on (ℝ^d in the metric L_p((ℝ^d).     

用指数型整函数的最佳限制逼近

我们研究了由仅有实零点的代数多项式导出的微分算子确定的广义Sobolev类利用指数型整函数作为逼近工具的最佳限制逼近问题.利用Fourier变换和周期化等方法,得到在L_2(R)范数下的广义Sobolev光滑函数类的相对平均宽度和最佳限制逼近的精确常数,以及当0是这个代数多项式的一个至多2重的零点时,得到最佳限制逼近在L_1(R)范数和一致范数下的广义Sobolev类的精确到阶的结果.     

各向异性Sobolev及Besov类的平均单边宽度

给出了各向异性Sobolev类及各向异性Besov类的平均单边宽度,获得了相应的弱渐近估计.     

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.     

Widths between the anisotropic spaces and the spaces of functions with mixed smoothness

The Kolmogorov, Gelfand, linear and orthoprojection widths of the classes of functions with mixed smoothness in the anisotropic spaces and those of the anisotropic classes in the spaces of functions with mixed smoothness are considered. The optimal asymptotic order of the widths are given.     

AVERAGE ONESIDED WIDTHS OF SOBOLEV AND BESOV CLASSES

The article concerns the average onesided widths of the Sobolev and Besov classes and the classes of functions with bounded moduli of smoothness. The weak asymptotic results are obtained for the corresponding quantities.