On the Best Least Squares Approximation of Continuous Functions using Linear Splines with Free Knots

Approximations to continuous functions by linear splines cangenerally be greatly improved if the knot points are free variables.In this paper we address the problem of computing a best linearspline L₂-approximant to a given continuous function on a givenclosed real interval with a fixed number of free knots. We describe an algorithm that is currently available and establishthe theoretical basis for two new algorithms that we have developedand tested. We show that one of these new algorithms had goodlocal convergence properties by comparison with the other techniques,though its convergence is quite slow. The second new algorithmis not so robust but is quicker and so is used to aid efficiency.A starting procedure based on a dynamic programming approachis introduced to give more reliable global convergence properties. We thus propose a hybrid algorithm which is both robust andreasonably efficient for this problem.     

Common Fixed Points and Best Approximation

We prove a common fixed-point theorem generalizing results of Dotson and Habiniak. Using this result, we extend, generalize, and unify well known results on fixed points and common fixed points of best approximation.     

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．     

Critical Points and Error Rank in Best H 2 Matrix Rational Approximation of Fixed McMillan Degree

This paper deals with best rational approximation of prescribed McMillan degree to matrix-valued functions in the real Hardy space of the complement of the unit disk endowed with the Frobenius L ₂ -norm. We describe the topological structure of the set of approximants in terms of inner-unstable factorizations. This allows us to establish a two-sided tangential interpolation equation for the critical points of the criterion, and to prove that the rank of the error F-H is at most k-n when F is rational of degree k , and H is critical of degree n . In the particular case where k=n , it follows that H=F is the unique critical point, and this entails a local uniqueness result when approximating near-rational functions. January 23, 1996. Date revised: September 16, 1996.     

对没有凸值和连续性的集值映象的最佳逼近定理*

本文引入和研究了一类具有弱凸图的集值映象.对这类映象证明了某些新的重合定理,最佳逼近和不动点定理.     

Inequalities for the Extended Best Polynomial Approximation Operator in Orlicz Spaces

In this paper we pursue the study of the best approximation operator extended from L~Φ to L~φ, where φ denotes the derivative of the function Φ. We get pointwise convergence for the coefficients of the extended best approximation polynomials for a wide class of function f, closely related to the Calder′on–Zygmund class t_m~p(x) which had been introduced in 1961. We also obtain weak and strong type inequalities for a maximal operator related to the extended best polynomial approximation and a norm convergence result for the coefficients is derived. In most of these results, we have to consider Matuszewska–Orlicz indices for the function φ.     

单纯形上的Stancu多项式与最佳多项式逼近

作为Bernstein多项式的推广,本文定义单纯形上的多元Stancu多项式.以最佳多项式逼近为度量,建立Stancu多项式对连续函数的逼近定理与逼近阶估计,给出Stancu多项式的一个逼近逆定理,从而用最佳多项式逼近刻划Stancu多项式的逼近特征.     

Zeros of the Jones Polynomial for Torus Knots and 2-bridge Knots

We study zeros of the Jones polynomial and their distributions for torus knots and 2-bridge knots. We prove that e(2m+1)πi/2and e(2m+1)πi/4(m is a positive integer)can not be the zeros of Jones polynomial for torus knots T p,q by the knowledge of the trigonometric function. We elicit the normal form of Jones polynomials of the 2-bridge knot C(-2, 2, · · ·,(-1)r2) by the recursive form and discuss the distribution of their zeros.     

On Characterization of Best Approximation with Certain Constraints

SupposeKis the intersection of a finite number of closed half-spaces {K_i} in a Hilbert spaceX, andx∈X\K. Dykstra's cyclic projections algorithm is a known method to determine an approximate solution of the best approximation ofxfromK, which is denoted byP_K(x). Dykstra's algorithm reduces the problem to an iterative scheme which involves computing the best approximation from the individualK_i. It is known that the sequence {x_j} generated by Dykstra's method converges to the best approximationP_K(x). But since it is difficult to find the definite value of an upper bound of the error ‖x_j−P_K(x)‖, the applicability of the algorithm is restrictive. This paper introduces a new method, called thesuccessive approximate algorithm, by which one can generate a finite sequencex₀, x₁, …, x_kwithx_k=P_K(x). In addition, the error ‖x_j−P_K(x)‖ is monotone decreasing and has a definite upper bound easily to be determined. So the new algorithm is very applicable in practice.     

一类二次插值样条达到最优逼近阶的特征

Let △={0=x₀1<…x_n=1} be a partition of [0,1], and let h_i=x_i-x_i-1(i=1,…,n) and △=(?)h_i. The set of splines of degree 2 and deficiency 1 on △ is denoted by S₂.     

强CHIP性质和广义限制域逼近的特征

本文研究了广义限制域的最佳逼近问题,在允许有有限个节点的情况下,引入次强内点条件的概念,并将优化理论中的强CHIP性质等概念应用到本文所研究的问题中,刻画了次强内点、强CHIP性质和最佳逼近的特征之间的关系.作为推论,我们得到了广义限制最佳逼近的Kolmogorov型和“零属于凸包”型等特征定理.     

置换空间中的最佳逼近存在性与唯一性定理

得到了置换空间PXXn中的最佳逼近存在性与唯一性定理.     

Coincidence Theorems with Applications to Minimax Inequalities, Section Theorem, Best Approximation and Multiobjective Games in Topological Spaces

Some new coincidence theorems involving admissible set-valued mappings are proved in general noncompact topological spaces. As applications, some new minimax inequalities, section theorem, best approximation theorem, existence theorems of weighted Nash equilibria and Pareto equilibria for multiobjective games are given in general topological spaces.     

BCQ条件和广义限制域最佳一致逼近的特征

研究了广义限制域的最佳一致逼近问题,在允许有有限个节点的情况下,引入次强内点条件的概念,并将优化理论中的BCQ条件等概念应用到本文所研究的问题中,刻划了次强内点条件、BCQ条件和最佳一致逼近的特征之间的关系.     

A Characterization of Best φ-Approximants with Applications to Multidimensional Isotonic Approximation

Some properties of best monotone approximants in several variables are obtained. We prove the following abstract characterization theorem. Let $(\om, {\cal A},\mu)$ be a measurable space and let ${\cal L}\subset{\cal A}$ be a $\sigma$-lattice. If $f$ belongs to a Musielak–Orlicz space $L_{\varphi}(\Omega, {\cal A},\mu),$ then there exists a $\sigma$-algebra ${\cal A}_f\subset{\cal A}$ such that $g$ is a best $\varphi$-approximant to $f$ from $L_{\varphi}({\cal L})$ iff $g$ is a best $\varphi$-approximant to $f$ from $L_{\p}({\cal A}_f)$. The $\sigma$-algebra ${\cal A}_f$ depends only on $f$. When $\Omega\subset\mbox{{\bf R}}^n$ and $L_{\varphip}({\cal L})$ is the set of monotone functions in several variables, we give sufficient conditions on the geometry of $\Omega$ to obtain a uniqueness theorem. This result extends and unifies previous ones. Finally, we prove a coincidence relation between a function and its best $\varphi$-approximant. Our main results are new, even in the classical Lebesgue spaces $L_p$.     

范德蒙德行列式与多项式的不动点

以范德蒙德行列式为工具,得到关于多项式不动点的两个新结论,以及线性代数中某已知结论的一个新证明,另外提供实例说明多项式不动点的应用．