W_2~m空间中样条插值算子与最佳逼近算子的一致性

由线性微分算子确定的样条是连接多项式样条与希氏空间中抽象算子样条的重要环节,对微分算子样条的研究,既可从更高的观点揭示和概括多项式样条,又可启示我们去发现抽象算子样条的一些新的理论和应用． Ｇｒｅｅｎ函数是研究微分算子样条的重要工具 ［１］,但在微分算子插值样条的计算及将样条用于数值分析中,再生核方法起着更重要的作用．文献［２］［３］给出了与二阶线性微分算子插值样条有关的再生核解析表达式;由此得到了二阶微分算子插值样条与空间Ｗ_２～１［ａ,ｂ］中最佳插值逼近算子的一致性;而且还利用再生核给出了Ｈｉ…     

W_2~m空间中样条插值算子与最佳逼近算子的一致性

This paper discusses generalized interpolating splines which determined by n order linear differential operators, and the best operators of interpolating approximation in W_2~m spaces, The explicit constructive method for the reproducing kernel in W_2~m space is presented, and proves the uniformity of spline interpolating operators and the best operators of interpolating approximation W_2~m space by reproducing kernel. The explicit expression of approximation error on a bounded ball in W_2~m space, and error estimation of spline operator of approximation are obtained.     

Small sets in best approximation theory

The best approximation problem to a nonempty closed set in a locally uniformly convex Banach space is considered. The main result states that the set of points which have best approximation but the approximation problem is not well-posed is very small in a sense that it is σ-cone supported in the underlying space. This gives an improvement of an original result of Stečkin about the set of points with more than one best approximation which involves Baire categories. Examples on the necessity of some of the imposed conditions are provided.     

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.     

Chebyshev centers proximinality and farthest points in strong normed almost linear spaces

Some results from the theory of best (or best simultaneous) approximation in a normed linear space have been extended to a normed almost linear space [strong normed almost linear space].     

Banach 空间中非线性最佳逼近的广义强唯一性

设X是一致凸空间,G为X中太阳集,R．Smarzewski[1]证明了g∈G对x∈的最佳逼近具有广义强唯一性,本文讨论其逆,在最佳逼近是广义强唯一的条件下,研究了空间的凸性和逼近集的太阳性．     

Geometry and Topology of Continuous Best and Near Best Approximations

The existence of a continuous best approximation or of near best approximations of a strictly convex space by a subset is shown to imply uniqueness of the best approximation under various assumptions on the approximating subset. For more general spaces, when continuous best or near best approximations exist, the set of best approximants to any given element is shown to satisfy connectivity and radius constraints.     

赋β-范空间中的最佳逼近问题

This paper deals with the problems of best approximation in β-normed spaces.With the tool of conjugate cone introduced in [1] and via the Hahn-Banach extension theorem of β-subseminorm in [2],the characteristics that an element in a closed subspace is the best approximation are given in Section 2.It is obtained in Section 3 that all convex sets or subspaces of a β-normed space are semi-Chebyshev if and only if the space is itself strictly convex.The fact that every finite dimensional subspace of a strictly convex β-normed space must be Chebyshev is proved at last.     

Proximity maps, best approximations and fixed points

In [Canada Math. Bull. 22, pp. 513–515, 1979], the author extended an interesting (best approximation) theorem of Ky Fan [see Theorem 2, Math. Z. 112, pp. 234–240] to condensing maps defined on a closed ball in a Banach space and defined on a closed convex subset in a Hilbert space. In [J. Approximation Theory 52, pp. 141–148, 1988], the author and Yen further extended it to 1-set-contractive maps in a Hilbert space. In this paper, we will address several related questions. First, we extend Lin and Yens’ result for 1-set-contractive maps but to satisfy weaker condition, which allows us to include generalized contraction maps. Second, we give ways to construct the best approximation (not just the existence results) for nonexpansive maps. Third, we give conditions on when the best approximation {u_n|n=1,2,…} of a sequence of contractions {f_n|n=1,2,…} will converge to the best approximation u of a contraction f. Fourth, common best approximation theorems for nonexpansive maps are proved. Applications to fixed point theorems for weakly inward maps and others are given for all those four topics.     

Proximity maps, best approximations and fixed points

In [Canada Math. Bull. 22, pp. 513–515, 1979], the author extended an interesting (best approximation) theorem of Ky Fan [see Theorem 2, Math. Z. 112, pp. 234–240] to condensing maps defined on a closed ball in a Banach space and defined on a closed convex subset in a Hilbert space. In [J. Approximation Theory 52, pp. 141–148, 1988], the author and Yen further extended it to 1-set-contractive maps in a Hilbert space. In this paper, we will address several related questions. First, we extend Lin and Yens’ result for 1-set-contractive maps but to satisfy weaker condition, which allows us to include generalized contraction maps. Second, we give ways to construct the best approximation (not just the existence results) for nonexpansive maps. Third, we give conditions on when the best approximation {u_n|n=1,2,…} of a sequence of contractions {f_n|n=1,2,…} will converge to the best approximation u of a contraction f. Fourth, common best approximation theorems for nonexpansive maps are proved. Applications to fixed point theorems for weakly inward maps and others are given for all those four topics.     

On best approximation by rational and holomorphic mappings between Banach spaces

After proving a generalized version of Garkavi's theorem, we give as applications proofs of existence results on best approximation by polynomials, and fractional linear and holomorphic operators between Banach spaces. We also obtain theorems on best approximation by some types of rational functions defined in open subsets of Banach spaces. By considering a natural non-normable distance we prove that every mapping bounded on the bounded subsets of a Banach space has best approximation by polynomials of degree less than or equal to a fixed natural number n.     

联合逼近的存在性与离散化

本文研究了紧距离空间上的函数集合被满足Young条件的函数类的联合逼近问题,建立了存在性定理,得到了极限定理,证明了可以用函数集合的离散化或距离空间的离散化得到最佳联合逼近。     

赋$\beta$-范空间中的最佳逼近问题

本文讨论赋$\beta$-范空间中的最佳逼近问题.以[1]引进的共轭锥为工具,借助[2]中关于$\beta$-次半范的Hahn-Banach延拓定理,第二节给出赋$\beta$-范空间的闭子空间中最佳逼近元的特征,第三节得到赋$\beta$-范空间中任何凸子集或子空间均为半Chebyshev集的充要条件是空间本身严格凸,文章最后证明了严格凸的赋$\beta$-范空间中任何有限维子空间都是Chebyshev集.     

A characterization of an element of best simultaneous approximation

Deutsch [4] has suggested that some problems of best simultaneous approximation might profitably be viewed as problems of best approximation in an appropriate product space. A few authors have touched upon this approach; none, however, have pursued it consistently or developed a complete problem along such a line, even in the simplest of cases. In this paper, we show that Deutsch's suggestion can easily be carried out using known results from approximation theory to establish existence, uniqueness, and characterization results. An algorithm guaranteed to converge strongly to the element of best simultaneous approximation under certain circumstances is also proposed.     

关于非线性共同逼近的强唯一性

本文首先指出文献［１］中的一个错误，举例说明弱拟凸集的最佳逼近未必具有广义强唯一性，进而讨论两类共同逼近的强唯一性，在空间是一致凸、逼近集是共同太阳集的条件下，证明了最佳共同逼近具有广义强唯一性     

On vector valued function approximation using a peak norm

A peak norm is defined for L_p spaces of E-valued Bochner integrable functions, where E is a Banach space, and best approximations from a sun to elements of the space are characterized. Applications are given to some families of simultaneous best approximation problems.     

The equivalence relations between the best approximation and linear operators approximation in a Besov space

In this paper, we characterize Besov type space introduced by the best approximation, best approximating elements and a kind of linear operators. The project is supported by NSFC     

Best Simultaneous Approximation and Its Applications in C ℝ(Q)

We develop a theory of best simultaneous approximation for closed convex sets in C _?(Q), the space of all real-valued continuous functions on a compact topological space Q endowed with the usual operations and with the norm ‖x‖ = max _q?Q |x(q)|. We give necessary and sufficient conditions for the existence of best simultaneous approximation in a conditionally complete Banach lattice X with a strong unit 1 by elements of the hyperplanes. We study best simultaneous approximation by elements of closed convex sets in C _?(Q) and give various characterizations of best simultaneous approximation.     

The similarity between the best approximation and the best copositive approximation

In this paper the author studies the copositive approximation in C(?) by elements of finite dimensional Chebyshev subspaces in the general case when ? is any totally ordered compact space. He studies the similarity between me behavior of the ordinary best approximation and the behavior pf the copositive best approximation. At the end of this paper, the author isolates many cases at which the two behaviors are the same.