Hermite-Birkhoff Interpolation on Arbitrarily Distributed Data in Banach Spaces

A class of cardinal basis functions is proposed in order to achieve a generalization to Banach spaces of Hermite-Birkhoff interpolation on arbitrarily distributed data. First, a constructive characterization of the class of cardinal basis functions is given. Then, the interpolation problem is solved by using a suitable combination of such functions and Taylor-Fréchet expansions. The performance of the obtained interpolants is improved by applying a localizing scheme, and the corresponding approximation error is estimated. A noteworthy case in Hilbert spaces and a numerical test comparing the Hermite-Birkhoff and Lagrange interpolants complete the presentation.     

Lagrange Interpolation in Weighted Besov Spaces

The behavior of the Lagrange polynomial L _m (w,f) , based on the zeros of the orthogonal polynomials, is studied in some weighted Besov spaces B ^p _r,q (u) . It is proved that L _m (w) is a uniformly bounded map under suitable conditions on the weight functions and the parameters p , r , and q . December 11, 1996. Date revised: October 29, 1997. Date accepted: June 15, 1998.     

On the existence and nonexistence of Lagrange multipliers in Banach spaces

The paper deals with the existence of Lagrange multipliers for a general nonlinear programming problem. Some regularity conditions are formulated which are, in a sense, the weakest to assure the existence of multipliers. A number of related conditions are discussed. The connection between the choice of suitable function spaces and the existence of multipliers is analyzed.This work was partly supported by the National Science Foundation, Grant No. GF-37298, to the Institute of Automatic Control, Technical University of Warsaw, Warsaw, Poland, and the Department of Computer and Control Sciences, University of Minnesota, Minneapolis, Minnesota.The author wishes to thank Professor A. P. Wierzbicki for many important remarks concerning the subject of this paper.     

多元分次Lagrange插值

对多元多项式分次插值适定结点组的构造理论进行了深入的研究与探讨.在沿无重复分量代数曲线进行Lagrange插值的基础上,给出了沿无重复分量分次代数曲线进行分次Lagrane插值的方法,并利用这一结果进一步给出了在R~2上构造分次Lagrange插值适定结点组的基本方法.另外,利用弱Gr(o|¨)bner基这一新的数学概念,以及构造平面代数曲线上插值适定结点组的理论,进一步给出了构造平面分次代数曲线上分次插值适定结点组的方法,从而基本上弄清了多元分次Lagrange插值适定结点组的几何结构和基本特征.     

Interpolation for Function Spaces Related to Mixed Boundary Value Problems

Interpolation theorems are proved for Sobolev spaces of functions on nonsmooth domains with vanishing trace on a part of the boundary.     

Weighted Convergence of Lagrange Interpolation Based on Gauss-Kronrod Nodes

The Gauss-Kronrod quadrature scheme, which is based on the zeros of Legendrepolynomials and Stieltjes polynomials, is a standard rule for automaticnumerical integration in mathematical software libraries. For a long time,very little was known about the underlying Lagrange interpolationprocesses. Recently, the authors proved new bounds and asymptoticproperties for the Stieltjes polynomials and, subsequently, appliedthese results to investigate the associated interpolation processes. Thepurpose of this paper is to survey the quality of these interpolationprocesses, with additional results that extend and complete the existingones. The principal new results in this paper are necessary and sufficientconditions for weighted convergence. In particular, we show that theLagrange interpolation polynomials associated with the above interpolationprocesses have the same speed of convergence as the polynomials of bestapproximation in certain weighted Besov spaces.     

Lagrange插值公式与Hermite插值公式的注记

给出一种基于商的形式的Lagrange与Hermite插值公式及其证明,同时还给出了两个相关的不等式.     

Lagrange Interpolation on a Sphere

§ 1.Ｉｎｔｒｏｄｕｃｔｉｏｎ　ＬｅｔｎｂｅａｎｏｎｎｅｇａｔｉｖｅｉｎｔｅｇｅｒａｎｄＳ ={(ｘ ,ｙ ,ｚ)∈Ｒ3 |ｘ2 + ｙ2 +ｚ2 =1 }ｂｅｔｈｅｕｎｉｔｓｐｈｅｒｅｉｎＲ3 .Ｐ( 2 )ｎ ａｎｄＰ( 3 )ｎ ｄｅｎｏｔｅｔｈｅｓｐａｃｅｏｆａｌｌｂｉｖａｒｉａｔｅｐｏｌｙｎｏｍｉａｌｓｏｆｔｏｔａｌｄｅｇｒｅｅ≤ｎａｎｄｔｈｅｓｐａｃｅｏｆａｌｌｔｒｉｖａｒｉａｔｅｐｏｌｙｎｏｍｉａｌｓｏｆｔｏｔａｌｄｅｇｒｅｅ≤ｎｒｅｓｐｅｃｔｉｖｅｌｙ ,ｉ.ｅ .Ｐ( 2 )ｎ =∑0≤ｉ+ｊ≤ｎａｉｊｘｉｙｊ｜ａｉｊ ∈Ｒ ,Ｐ( 3 …     

插值法在数据修正中的应用

为了使评估的结果达到某种规定的水平,本文研究了运用线性插值、拉格朗日插值以及牛顿插值方法对某公司员工考核数据按照一定的规则进行了修正,同时,对各种方法的修正前、后的结果做了比较．结果表明拉格朗日插值法效果最好,但是计算量偏大;线性插值法虽然效果一般,但是计算复杂度却较低;而牛顿插值法达不到我们预期的效果．     

球面上的Lagrange插值

In this paper, we obtain a properly posed set of nodes for interpolation on a sphere. Moreover it is applied to construct properly posed set of nodes for Lagrange interpolation on the trivariate polynomial space of total degree n.     

Linear Optimization with Box Constraints in Banach Spaces

Let X be a partially ordered real Banach space, let a,b∈X with a≤b. Let φ be a bounded linear functional on X. We say that X satisfies the box-optimization property (or X is a BOP space) if the box-constrained linear program: max 〈φ,x〉, s.t. a≤x≤b, has an optimal solution for any φ,a and b. Such problems arise naturally in solving a class of problems known as interval linear programs. BOP spaces were introduced (in a different language) and systematically studied in the first author’s doctoral thesis. In this paper, we identify new classes of Banach spaces that are BOP spaces. We present also sufficient conditions under which answers are in the affirmative for the following questions:

1. (i)
   When is a closed subspace of a BOP space a BOP space?
    
2. (ii)
   When is the range of a bounded linear map a BOP space?
    
3. (iii)
   Is the quotient space of a BOP space a BOP space?
    

     

Banach空间中的β扰动优化

通过在Banach空间中引入一种新拓扑来证明其上的β扰动优化定理成立。     

A Modification of Lagrange Interpolation

A modification of Lagrange interpolation based on the zeros of the Chebyshev polynomial of the second kind is constructed, which interpolates at many ofgiven data. Thus, for this node-system the main result gives an affimative answer to a problem suggested by Bernstein in 1930. Moreover, our modification has a Timan-Gopengauz type approximation rate.     

K-Theory for the Banach Algebra of Operators on James''s Quasi-reflexive Banach Spaces

We prove that the K-groups of the Banach algebra of bounded, linear operators on the pth James space , where 1 < p < , are given by and . Moreover, for each Banach space and each non-zero, closed ideal contained in the ideal of inessential operators, we show that and . This enables us to calculate the K-groups of for each Banach space which is a direct sum of finitely many James spaces and -spaces.     

Interpolation properties of scales of Banach spaces

It is shown that any interpolation scales joining weight spaces L _p or similar spaces have many remarkable properties. Not only are such scales intrinsically interpolation scales, but an analog of the Arazy-Cwikel theorem describing interpolation spaces between the spaces from the scale is valid.     

内插空间理论的应用

综述了线性算子内插法与内插空间理论在Banach空间几何学,微分算子,逼近理论,积分算子,Fourier分析等领域的一些应用。     

Interpolation of Banach algebras and tensor products of Banach couples

Using tensor products of Banach couples we study a class of interpolation functors with the property that to every Banach couple of Banach algebras they give an interpolation space which is a Banach algebra. For the real θ,1-method we give a complete answer to the question of when the interpolation space is unital.     

Banach空间的K—M逼近

设Ｘ是自反、严格凸且具有Ｈ性质的Ｂａｎａｃｈ空间，Ａ是Ｘ的弱序列完备子集。本文证明了Ａ是可逼近紧的Ｃｈｅｂｙｓｈｅｖ集的充分必要条件是Ａ是太阳集；同时，也讨论了Ｏｒｌｉｃｚ序列空间的太阳集。     

Lagrange插值和Hermite-Fejér插值在Wiener空间下的平均误差

在L_q-范数逼近的意义下,确定了基于Chebyshev多项式零点的Lagrange插值多项式列和Hermite-Fejér插值多项式列在Wiener空间下的p-平均误差的弱渐近阶.从我们的结果可以看出,当2≤q<∞,1≤p<∞时,基于第一类Chebyshev多项式零点的Lagrange插值多项式列和Hermite-Fejér插值多项式列的p-平均误差弱等价于相应的最佳逼近多项式列的p-平均误差.在信息基计算复杂性的意义下,如果可允许信息泛函为计算函数在固定点的值,那么当1≤p,q<∞时,基于第一类Chebyshev多项式零点的Lagrange插值多项式列和Hermite-Fejér插值多项式列在Wiener空间下的p-平均误差弱等价于相应的最小非自适应p-平均信息半径.     

Gradient Estimation Using Lagrange Interpolation Polynomials

We use Lagrange interpolation polynomials to obtain good gradient estimations. This is e.g. important for nonlinear programming solvers. As an error criterion, we take the mean squared error, which can be split up into a deterministic error and a stochastic error. We analyze these errors using N-times replicated Lagrange interpolation polynomials. We show that the mean squared error is of order if we replicate the Lagrange estimation procedure N times and use 2d evaluations in each replicate. As a result, the order of the mean squared error converges to N ⁻¹ if the number of evaluation points increases to infinity. Moreover, we show that our approach is also useful for deterministic functions in which numerical errors are involved. We provide also an optimal division between the number of gridpoints and replicates in case the number of evaluations is fixed. Further, it is shown that the estimation of the derivatives is more robust when the number of evaluation points is increased. Finally, test results show the practical use of the proposed method. We thank Jack Kleijnen, Gül Gürkan, and Peter Glynn for useful remarks on an earlier version of this paper. We thank Henk Norde for the proof of Lemma 2.2.