WEIGHTED BEST LOCAL APPROXIMATION IN ORLICZ SPACE

We present a common fixed point theorem for generalized asymptotically non- expansive and noncommuting mappings in normed linear spaces.     

ON WEIGHTED L~p INTEGRABILITY AND APPROXIMATION OF TRIGONOMETRIC SERIES

We generalize several classical results on the integrability of trigonometric series and relations among the best approximation and the coefficients of trigonometric series. Theorem 3 and Theorem 4 are the first results on the relations among the weighted best approximation and the coefficients of trigonometric series.     

Best L2 Local Approximation on Two Small Intervals

In this article, we introduce the τ condition, which is weaker than the L² differentiability. If a function satisfies the τ condition on two points of ?, we prove the existence and characterization of the best local polynomial approximation on these points.     

Weighted inequalities and applications to best local approximation in Luxemburg norm

In this paper we establish a result about uniformly equivalent norms and the convergence of best approximant pairs on the unitary ball for a family of weighted Luxemburg norms with normalized weight functions depending on ε, when ε→ 0. It is introduced a general concept of Pade approximant and we study its relation with the best local quasi-rational approximant. We characterize the limit of the error for polynomial approximation. We also obtain a new condition over a weight function in order to obtain inequalities in Lp norm, which play an important role in problems of weighted best local Lp approximation in several variables.     

Weighted Best Local Approximation

In this survey, the notion of a balanced best multipoint local approximation is fully exposed since they were treated in the Lpspaces and recent results in Orlicz spaces. The notion of balanced point, which was introduced by Chui et al. in 1984 are extensively used.     

由Fourier系数确定的函数类的最佳m-项单边逼近渐近估计

结合最佳m项逼近和单边逼近的思想引进所谓最佳m项单边逼近的概念,给出由Fourier系数确定的光滑函数类通过三角函数系在Lp(1≤P≤∞)的最佳m-项单边逼近渐近估计以及m-项类贪婪单边逼近结果.     

Weak Type Inequalities for Best Simultaneous Approximation in Banach Spaces

For arbitrary Banach spaces Butzer and Scherer in 1968 showed that the approximation order of best approximation can characterized by the order of certain K-functionals. This general theorem has many applications such as the characterization of the best approximation of algebraic polynomials by moduli of smoothness involving the Legendre, Chebyshev, or more general the Jacobi transform. In this paper we introduce a family of seminorms on the underlying approximation space which leads to a generalization of the Butzer–Scherer theorems. Now the characterization of the weighted best algebraic approximation in terms of the so-called main part modulus of Ditzian and Totik is included in our frame as another particular application. The goal of the paper is to show that for the characterization of the orders of best approximation, simultaneous approximation (in different spaces), reduction theorems, and K-functionals one has (essentially) only to verify three types of inequalities, namely inequalities of Jackson-, Bernstein-type and an equivalence condition which guarantees the equivalence of the seminorm and the underlying norm on certain subspaces. All the results are given in weak-type estimates for almost arbitrary approximation orders, the proofs use only functional analytic methods.     

赋Luxemburg范数的Musielak-Orlicz-Sobolev空间单调性在最佳逼近中的应用

借鉴Orlicz-Sobolev空间的单调性在最佳逼近中一些应用,利用Musielak-Orlicz-Sobolev空间的构成特点,给出了赋Luxemburg范数的Musielak-Orlicz-Sobolev空间的单调性在最佳逼近中一些应用的充分性定理.     

On weighted L p integrability and approximation of trigonometric series

We generalize several classical results on the integrability of trigonometric series and relations among the best approximation and the coefficients of trigonometric series. Theorem 3 and Theorem 4 are the first results on the relations among the weighted best approximation and the coefficients of trigonometric series. Supported in part by Hangzhou Normal University.     

单隐层神经网络与最佳多项式逼近

研究单隐层神经网络逼近问题．以最佳多项式逼近为度量,用构造性方法估计单隐层神经网络逼近连续函数的速度．所获结果表明:对定义在紧集上的任何连续函数,均可以构造一个单隐层神经网络逼近该函数,并且其逼近速度不超过该函数的最佳多项式逼近的二倍．     

用子空间FT2n-1⊕SΔrN 逼近函数类Wrp 的最佳逼近度与其K-宽度之间的渐近关系

本文研究了用子空间FT2n-1⊕SΔrN 逼近函数类Wrp的最佳逼近度与其K-宽度之间的关系.利用Kolmogorov-宽度的概念和Kolmogorov比较定理,获得了其最佳逼近度与其Kolmogorov-宽度之间的渐进性质,推广了用子空间FT2n-1⊕SΔrN 逼近函数类Wrp 的最佳逼近度的有关结论.     

复赋范空间中复RS集的最佳同时逼近

研究了复赋范空间中具限制系数的广义多项式集G对无穷序列的最佳同时逼近问题,得到了特征定理;当G是复RS集时还得到了惟一性定理．     

Jackson inequality for Banach spaces on the sphere

The best rate of approximation of functions on the sphere by spherical polynomials is majorized by recently introduced moduli of smoothness. The treatment applies to a wide class of Banach spaces of functions.      

Best m-term one-sided trigonometric approximation of some function classes defined by a kind of multipliers

In this paper, we continue studying the so-called non-linear best m-term one-sided approximation problems and obtain the asymptotic estimations of non-linear best m-term one-sided trigonometric approximation under the norm Lp （1 ≤ p ≤ ∞） of multiplier function classes and the corresponding m-term Greedy-liked one-sided trigonometric approximation results.     

基于块循环矩阵的对称张量的最佳秩-1逼近

对称张量的最佳秩-1问题是张量研究中非常重要的部分.首先,基于三阶张量的块循环矩阵,提出了求解对称张量最佳秩-1逼近问题的一个新方法.其次,针对求解对称张量的最佳秩-1逼近方法,给出了对称张量的最佳秩-1逼近不变性的一个充要条件,以及逼近误差上界的估计.最后,数值算例表明了上述方法的可行性和误差上界的正确性.     

Banach空间中不适定线性算子方程的最佳逼近解

设X,Y为Banach空间,T为从X到Y的线性算子.T的值域R(T)≠Y且为逼近紧子空间,T的零空间N(T)≠{θ}.证得不适定算子方程Tx=y的最佳逼近解对任意y∈Y均存在的充分必要条件是N(T)为X的迫近子空间.     

Best Local Approximation in Orlicz Spaces

We get results in Orlicz spaces L ^φ about best local approximation on non-balanced neighborhoods when φ satisfies a certain asymptotic condition. This fact generalizes known previous results in L ^p spaces.     

On well posedness of best simultaneous approximation problems in Banach spaces

The well posedness of best simultaneous approximation problems is considered. We establish the generic results on the well posedness of the best simultaneous approximation problems for any closed weakly compact nonempty subset in a strictly convex Kadec Banach space. Further, we prove that the set of all points inE(G) such that the best simultaneous approximation problems are not well posed is a u- porous set inE(G) whenX is a uniformly convex Banach space. In addition, we also investigate the generic property of the ambiguous loci of the best simultaneous approximation.     

多元Bernstein-Durrmeyer型多项式及其逼近特征

作为Bernstein-Durrmeyer多项式的推广,定义单纯形上的Bernstein-Durrmeyer型多项式.以最佳多项式逼近为度量,给出Bernstein-Durrmeyer型多项式Lp逼近阶的估计,并且以一个逆向不等式的形式建立其Lp逼近的逆定理,从而用最佳多项式逼近刻画该多项式Lp逼近的特征.所获结果包含了多元Bernstein-Durrmeyer多项式的相应结果.     

Method of Best Successive Approximations for Nonlinear Operators

We present a new approach to the approximation of nonlinear operators in probability spaces. The approach is based on a combination of the specific iterative procedure and the best approximation problem solution with a quadratic approximant. We show that the combination of these new techniques allow us to build a computationally efficient and flexible method. The algorithm of the method and its application to the optimal filtering of stochastic signals are given.