用样条逼近Sobolev函数类W_p~1(R)的逼近误差

作者研究了定义在全实轴上的Sobolev函数类W_p~1(R)的逼近问题.以一次样条函数作为逼近工具,给出了p=1和p=∞时的逼近误差.     

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.     

具有Gauss测度的Sobolev空间上的函数逼近

本文讨论了具有Gauss测度的Sobolev空间上的一元周期函数被三角多项式子空间的最佳逼近及被Fourier部分和算子,Vallée—Poussin算子,Ceshxo算子,Abel算子和Jackson算子的逼近,得到了平均误差估计．证明了在平均框架下,在Lq（1≤q〈∞）空间尺度下三角多项式子空间是渐进最优的子空间,但是在L∞空间尺度下,三角多项式子空间不是渐进最优的子空间．还证明了,Fourier部分和算子和Vallée-Poussin算子在Lq（1≤q≤∞）空间尺度下是渐进最优的线性算子．注意到在平均框架以及Lq（1≤q〈∞）空间尺度下,渐进最优的线性算子,如Fourier部分和算子及Vallée—Poussin算子,与最优的非线性算子的逼近效果一样好．     

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。     

确定性与随机化框架下的Sobolev类上的嵌入与积分的复杂性

本文研究各向异性Sobolev类上的嵌入以及积分问题的复杂性.我们得到这些问题在确定性、随机化框架以及平均框架下n-重最小误差的精确阶.所得结果表明在非嵌入连续函数空间情形,随机误差与平均误差实质性地小于确定性误差.从数量级看,对于嵌入问题,收敛阶最大改进可达到n-1+ε,这里ε是任意正数.对于积分问题最大改进可达到n...     

Sobolev圆盘代数上的乘法算子

研究一类由单位圆盘D上的Sobolev空间W^2,2(D)中的解析函数构成的代数, 称之为Sobolev圆盘代数, 给出了其上的有界线性乘法算子M_f的基本性质, 刻画了乘法算子M_f的换位子代数, 证明了A′(M_f)是交换的当且仅当M_f^*是指标为1的Cowen-Douglas算子.     

拟Hermite插值对解析函数类的逼近误差

在最大框架下研究基于第二类Tchebyshev节点组的拟Hermite插值算子和Hermite插值算子对一个解析函数类的逼近误差.对于一致范数,我们得到了相应量的精确值.对于L_p-范数(1≤p∞),我们得到了相应量的值或强渐近阶.     

多元Sobolev关于L_(qp)(R~d)下的平均线性宽度

得到了两种多元 Sobloev类于 Lqp( Rd )下平均线性宽度的弱渐进估计 .     

Hermite型多元样本定理及Sobolev类上混淆误差的估计

本文证明了Hermite型多元样本定理,并由此确定了Sobolev类上混淆误差阶的精确估计.     

Sobolev类带权函数的基于给定信息的最佳求积公式

给出了r阶Sobo lev类KWr[a,b]带权函数的基于给定信息的最佳求积公式和它的误差估计式.这里的给定信息是指:已知函数在给定区间若干点上的函数值和直到r-1阶导数值.对r≤2,得到了最佳求积公式和误差估计式的显式结果.另外还给出了类KW2[a,b]中在节点的导数值为零的函数所组成的子类的相应的最佳求积公式.     

Tractability of linear problems defined over Hilbert spaces

We study d$d$-variate approximation problems in the worst and average case settings. We consider algorithms that use finitely many evaluations of arbitrary linear functionals. In the worst case setting, we obtain necessary and sufficient conditions for quasi-polynomial tractability and uniform weak tractability. Furthermore, we give an estimate of the exponent of quasi-polynomial tractability which cannot be improved in general. In the average case setting, we obtain necessary and sufficient conditions for uniform weak tractability. As applications we discuss some examples.     

Approximation numbers of Sobolev embeddings—Sharp constants and tractability

We investigate optimal linear approximations (approximation numbers) in the context of periodic Sobolev spaces H^s(T^d)$H^s\left({\mathbb{T}}^d\right)$ of fractional smoothness s>0$s>0$ for various equivalent norms including the classical one. The error is always measured in _L2(T^d)$L_2\left({\mathbb{T}}^d\right)$. Particular emphasis is given to the dependence of all constants on the dimension d$d$. We capture the exact decay rate in n$n$ and the exact decay order of the constants with respect to d$d$, which is in fact polynomial. As a consequence we observe that none of our considered approximation problems suffers from the curse of dimensionality. Surprisingly, the square integrability of all weak derivatives up to order three (classical Sobolev norm) guarantees weak tractability of the associated multivariate approximation problem.     

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.     

Muntz Rational Approximation for Special Function Classes in Orlicz Spaces

Using the method of construction,with the help of inequalities,we research the Muntz rational approximation of two kinds of special function classes,and give the corresponding estimates of approximation rates of these classes under widely conditions.Because of the Orlicz Spaces is bigger than continuous function space and the Lp space,so the results of this paper has a certain expansion significance.     

Approximation by polynomials and smooth functions in Sobolev spaces with respect to measures

The density of polynomials is straightforward to prove in Sobolev spaces W^k,p((a,b)), but there exist only partial results in weighted Sobolev spaces; here we improve some of these theorems. The situation is more complicated in infinite intervals, even for weighted L^p spaces; besides, in the present paper we have proved some other results for weighted Sobolev spaces in infinite intervals.     

Sample Numbers and Optimal Lagrange Interpolation of Sobolev Spaces Wr1*

This paper investigates the optimal recovery of Sobolev spaces W^r₁[?1, 1], r ∈ N in the space L1[?1, 1]. They obtain the values of the sampling numbers of W^r₁[?1, 1] in L₁[?1, 1] and show that the Lagrange interpolation algorithms based on the extreme points of Chebyshev polynomials are optimal algorithms. Meanwhile, they prove that the extreme points of Chebyshev polynomials are optimal Lagrange interpolation nodes.