Best Approximation with Coefficient Constraints

For given = (₁,..., _n) and ß = (ß₁,...,ß_n), with – _i < ß_i (i = 1, ...,n) and continuous functions u₁,...,u_n, set This paper is concerned with best approximating continuous functions,in the uniform norm, from U(; ß). We exactly characterizethe u₁,..., u_n for which the best approximant to every continuousfunction is unique. We also present a general theorem characterizingall best approximants. When (u₁,..., u_n) is a Descartes, ora weak Descartes, system on [0, 1], explicit characterizationsof the best approximants in terms of equioscillations are given.These results are applied to spline spaces. They are also usedto complete the characterizations in certain specific examplespreviously considered in the literature.     

On the Problem of Best Rational Approximation with Interpolating Constraints (I)

The aim of this conjoint paper is to discuss the problem of the best rational approximation with interpolating constraints. In part I, we give the necessary and sufficient conditions for the existence of the best rational approximations and establish some characterization theorem for such approximations. The problems of uniqueness, the properties for the set of best approximations, strong uniqueness and continuity of best approximation operator are considered in Part II. The results obtained in this paper are the completion and extension of those given in [1].     

Best Approximation with Linear Constraints: A Model Example

We consider best approximation in L_p( ), 1 ≤ p ≤ ∞, by means of entire functions y of exponential type subject to additional constraints Γ_j(y) = 0, j = 1, ..., K. Here Γ_j are (unbounded) linear functionals of the form Γ_j(y) = Dⁿy(s_j) − ∑ a_kD^ky(s_j) where s_j are fixed points.     

Lagrange Multiplier Characterizations of Constrained Best Approximation with Infinite Constraints

Journal of Optimization Theory and Applications - In this paper, we first employ the subdifferential closedness condition and Guignard’s constraint qualification to present “dual cone...     

一类三角级数的最佳逼近度

本文在复值连续函数空间中给出一类具有O-正则变化拟单调系数之三角级数的最佳逼近度的一个完备结果.特别是第3节的定理3.2,即使在实连续函数空间中亦是一个新的结论,而且也是Xie与Zhou[6]的定理2在一个方向上的本质推广.最后,在第4节给出一个应用.     

On Best Simultaneous Approximation

The problem is considered of best approximation of finite number of functions simultaneously. For a very general class of norms, characterization results are derived. The main part of the paper is concerned with proving uniqueness and strong uniqueness theorems. For a particular subclass, which includes the important special case of the Chebyshev norm, a characterization is given of a uniqueness element.     

Best Approximation in Hardy Spaces and by Polynomials, with Norm Constraints

Two related approximation problems are formulated and solved in Hardy spaces of the disc and annulus. With practical applications in mind, truncated versions of these problems are analysed, where the solutions are chosen to lie in finite-dimensional spaces of polynomials or rational functions, and are expressed in terms of truncated Toeplitz operators. The results are illustrated by numerical examples. The work has applications in systems identification and in inverse problems for PDEs.     

On Approximation for Certain Operators

ＯｎＡｐｐｒｏｘｉｍａｔｉｏｎｆｏｒＣｅｒｔａｉｎＯｐｅｒａｔｏｒｓ￥（郭顺生）ＧｕｏＳｈｕｎｓｈｅｎｇ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＨｅｂｅｉＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ，Ｓｈｉｊｉａｚｈｕａｎｇ，０５００１６）Ａｂｓｔｒａ...     

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.     

Approximation by Partial Sums of Fourier Series and the Best Approximation of Certain Function Classes

Continuing investigations of S. A. Telyakovskii, S. B. Stechkin, A. I. Stepanets and R. M. Trigub, the author studies two questions about approximation of the class of periodic functions with a bounded -derivative by polynomials in the spaces C and L.First, the asymptotics of approximation by partial sums of Fourier series has been found without limitations for the decreasing order of . Second, the exact decreasing order of the best approximation of the class has been found. New examples are also given.     

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 $L_p$.     

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.     

On Certain Best Constants for Bernstein-Type Operators

We discuss the generalized version of a best-constant problem raised by Z. Li in a note which recently appeared in Journal of Approximation Theory. Some best constants for well known Bernstein-type operators are obtained.     

Best Approximation with Walsh Atoms

We consider the approximation in L ² R of a given function using finite linear combinations of Walsh atoms, which are Walsh functions localized to dyadic intervals, also called Haar—Walsh wavelet packets. It is shown that up to a constant factor, a linear combination of K atoms can be represented to relative error ɛ by a linear combination of orthogonal atoms. In finite dimension N, best approximation with K orthogonal atoms can be realized with an algorithm of order . A faster algorithm of order solves the problem with indirect control over K. Therefore the above result connects algorithmic and theoretical best approximation. Date received: July 6, 1995. Date revised: January 8, 1996.     

On the Zeros of Polynomials of Best Approximation

Given a function f, uniform limit of analytic polynomials on a compact, regular set E^N, we relate analytic extension properties of f to the location of the zeros of the best polynomial approximants to f in either the uniform norm on E or in appropriate L^q norms. These results give multivariable versions of one-variable results due to Blatt–Saff, Pleniak and Wójcik.     

关于四元数矩阵的最佳逼近问题

通过使用四元数矩阵的广义奇异值分解,给出了四元数矩阵最佳逼近问题‖AHXA-C‖2F+‖BHXB-D‖2F=min, s.t. XH=X的一般表达式.     

赋Orlicz范数的Orlicz空间中最佳逼近算子

在赋Ｏｒｌｉｃｚ范数的Ｏｒｌｉｃｚ空间中,给出最佳逼近算子单调性的一个充分条件和最佳逼近元存在定理．     

On the Element of Best Nonsymmetric Approximation in Spaces with Nonsymmetric Quasinorm

In this paper, we prove an existence theorem for the element of best nonsymmetric approximation in spaces with nonsymmetric quasinorm. Examples showing the importance of the conditions embodied in the theorem are presented. A criterion for determining the element of best nonsymmetric approximation is given.     

Estimate for the Best Approximation of Summable Functions of Several Variables with a Certain Symmetry of Fourier Coefficients

An upper bound for the best approximation of summable functions of several variables by trigonometric polynomials in the metric of L is determined in terms of Fourier coefficients. We consider functions representable by trigonometric series with certain symmetry of coefficients satisfying a multiple analog of the Sidon–Telyakovskii conditions.