近严格凸与最佳逼近

本文研究近严格凸与最佳逼近的关系．证明了Ｂａｎａｃｈ空间Ｘ是近严格凸的当且仅当Ｘ的每个子空间是紧－半－切比晓夫空间．     

Some extremal properties of multivariate polynomial splines in the metric Lp (Rd )

We constructed a kind of continuous multivariate spline operators as the approximation tools of the multivariate functions on the Bd instead of the usual multivariate cardinal interpolation oper-ators of splines, and obtained the approximation error by this kind of spline operators. Meantime, by the results, we also obtained that the spaces of multivariate polynomial splines are weakly asymptoti-cally optimal for the Kolmogorov widths and the linear widths of some anisotropic Sobolev classes of smooth functions on Bd in the metric Lp(Bd).     

Предельная связь для различных методов приближения периоди ческих функций сплай нами и тригонометрически ми полиномами

Asymptotic relation for linear, best, and best with restrictions, methods of approximation is studied for periodic functions, by subspaces of splines and trigonometric polynomials of fixed dimension. Estimates of deviation between approximations of fixed elements from a normed space by splines and polynomials are obtained.     

On best simultaneous approximation in quotient spaces

We assume that X is a normed linear space, W and M are subspaces of X. We develop a theory of best simultaneous approximation in quotient spaces and introduce equivalent assertions between the subspaces W and W ＋ M and the quotient space W/M.     

Some extremal properties of multivariate polynomial splines in the metric Lp(ℝd)

We constructed a kind of continuous multivariate spline operators as the approximation tools of the multivariate functions on the (?^d instead of the usual multivariate cardinal interpolation operators of splines, and obtained the approximation error by this kind of spline operators. Meantime, by the results, we also obtained that the spaces of multivariate polynomial splines are weakly asyrnptotically optimal for the Kolrnogorov widths and the linear widths of some anisotropic Sobolev classes of smooth functions on (?^d in the metric L_p((?^d).     

Some extremal properties of multivariate polynomial splines in the metric Lp(?d)

We constructed a kind of continuous multivariate spline operators as the approximation tools of the multivariate functions on the (ℝ^d instead of the usual multivariate cardinal interpolation operators of splines, and obtained the approximation error by this kind of spline operators. Meantime, by the results, we also obtained that the spaces of multivariate polynomial splines are weakly asyrnptotically optimal for the Kolrnogorov widths and the linear widths of some anisotropic Sobolev classes of smooth functions on (ℝ^d in the metric L_p((ℝ^d).     

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.     

Bivariate spline functions and the approximation of linear functionals

The reproducing kernel for a Hilbert space of bivariate functions which have Taylor expansions is constructed. The concepts of optimal approximation of linear functionals in the sense of Sard and approximations resulting from bivariate spline functions are shown to be equivalent in these spaces. Bivariate splines that both smooth and interpolate are discussed.This research was supported by the Office of Naval Research under Grant NR 044-443.     

Duality in quasi-convex supremization and reverse convex infimization via abstract convex analysis,and applications to approximation **

We give duality theorems and dual characterizations of optimal solutions for abstract quasi-convex supremization problems and infimization problems with abstract reverse convex constraint sets. Our main tools are dualities between families of subsets, conjugations of type Lau associated to them, and subdifferentials with respect to conjugations of type Lau. These tools permit us to give explicitly the relation between.the constraint sets, and the relation between the objective functions, of the primal problem and the dual problem. As applications, we obtain duality theorems for quasi-convex supremization and reverse convex infimization in locally convex spaces and, in particular, for worst and best approximation in normed linear spaces.     

How to build all Chebyshevian spline spaces good for geometric design?

In the present work we determine all Chebyshevian spline spaces good for geometric design. By Chebyshevian spline space we mean a space of splines with sections in different Extended Chebyshev spaces and with connection matrices at the knots. We say that such a spline space is good for design when it possesses blossoms. To justify the terminology, let us recall that, in this general framework, existence of blossoms (defined on a restricted set of tuples) makes it possible to develop all the classical geometric design algorithms for splines. Furthermore, existence of blossoms is equivalent to existence of a B-spline bases both in the spline space itself and in all other spline spaces derived from it by insertion of knots. We show that Chebyshevian spline spaces good for design can be described by linear piecewise differential operators associated with systems of piecewise weight functions, with respect to which the connection matrices are identity matrices. Many interesting consequences can be drawn from the latter characterisation: as an example, all Chebsyhevian spline spaces good for design can be built by means of integral recurrence relations.     

Approximation of curves by fairness splines with tangent conditions

We present in this paper an approximation method of curves from sets of Lagrangian data and vectorial tangent subspaces. We define a discrete smoothing fairness spline with tangent conditions by minimizing certain quadratic functional on finite element spaces. Convergence theorem is established and some numerical and graphical examples are analyzed in order to show the validity and the effectiveness of this paper.     

Best and coupled best approximation theorems in abstract convex metric spaces

In this paper, we first give a best approximation theorem in abstract convex metric spaces. As applications, we then derive some best and coupled best approximations and coupled coincidence point results in normed spaces and hyperconvex metric spaces.     

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.     

Banach空间的p— Asplund 伴随空间

我们称一个定义在Banach空间E上的连续凸函数f具有Frechet可微性质（FDP）,如果E上的每个实值凸函数g≤f均在E一个稠密的Gδ－子集上Frechet可微。本文主要证明了：对任何Banach空间E,均存在一个局部凸相容拓扑p使得1）（E,p)是Hausdorff局部凸空间;2） E上的每个范数连续具有FDP的凸函数均是p－连续的;3）每个p-连续的凸函数均具有FDP ;4）p等价某个范数拓扑当且仅不E是Asplund空间。     

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 class of multidimensional periodic splines

Periodic spline functions are introduced by use of reproducing kernel structure in Hilbert spaces. Minimum properties are described in interpolation and best approximation problems. A numerical method for determining interpolating splines and best approximations is proposed.Dedicated to Prof. Dr. F. Reutter on the occasion of his 70th birthday     

Non-separable splines and numerical computation of evolution equations by the Galerkin methods

We construct new non-separable splines and we apply the spline sampling approximation to the computation of numerical solutions of evolution equations. The non-separable splines are basis functions which give a fine sampling approximation which enables us to compute numerical solutions by means of the method of lines combined with the Galerkin method. To demonstrate our approach we compute numerical solutions of the Burgers equation and the Kadomtsev–Petviashvili equation.     

Integral representation of slowly growing equidistant splines

In this paper we consider polynomial splines S(x) with equidistant nodes which may grow as O (|x|^s). We present an integral representation of such splines with a distribution kernel. This representation is related to the Fourier integral of slowly growing functions. The part of the Fourier exponentials herewith play the so called exponential splines by Schoenberg. The integral representation provides a flexible tool for dealing with the growing equidistant splines. First, it allows us to construct a rich library of splines possessing the property that translations of any such spline form a basis of corresponding spline space. It is shown that any such spline is associated with a dual spline whose translations form a biorthogonal basis. As examples we present solutions of the problems of projection of a growing function onto spline spaces and of spline interpolation of a growing function. We derive formulas for approximate evaluation of splines projecting a function onto the spline space and establish therewith exact estimations of the approximation errors.     

On the Problem of Instability in the Dimensions of Spline Spaces over T-Meshes with T-Cycles

The T-meshes are local modification of rectangular meshes which allow T-junctions. The splines over T-meshes are involved in many fields, such as finite element methods, CAGD etc. The dimension of a spline space is a basic problem for the theories and applications of splines. However, the problem of determining the dimension of a spline space is difficult since it heavily depends on the geometric properties of the partition. In many cases, the dimension is unstable. In this paper, we study the instability in the dimensions of spline spaces over T-meshes by using the smoothing cofactor-conformality method. The modified dimension formulas of spline spaces over T-meshes with T-cycles are also presented. Moreover, some examples are given to illustrate the instability in the dimensions of the spline spaces over some special meshes.