The Bernstein Constant and Polynomial Interpolation at the Chebyshev Nodes

In this paper, we establish new asymptotic relations for the errors of approximation in L_p[−1,1], 0<p∞, of x^λ, λ>0, by the Lagrange interpolation polynomials at the Chebyshev nodes of the first and second kind. As a corollary, we show that the Bernstein constant

Full-size image

is finite for λ>0 and .     

THE DIVERGENCE OF LAGRANGE INTERPOLATION IN EQUIDISTANT NODES

It is a classical result of Bernstein that the sequence of Lagrange interpolation polynomials to [x] at equally spaced nodes in [- 1,1 ] diverges everywhere, except at zero and the end-points. In this paper we show that the sequence of Lagrange interpolation polynomials corresponding to the functions which possess better smoothness on equidistant nodes in [- 1,1 ] still diverges every where in the interval except at zero and the end-points.     

THE DIVERGENCE OF LAGRANGE INTERPOLATION FOR ｜x｜^α

This paper shows that the sequence of Lagrange interpolation polynomials corresponding to the rune tion f（z） =｜x｜^α（1〈α〈2） on [-1,1] can diverge everywhere in the interval except at zero and the end-points.     

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 …     

THE EXACT CONVERGENCE RATE AT ZERO OF LAGRANGE INTERPOLATION POLYNOMIAL TO ｜x｜^α

In this paper we present a generalized quantitative version of a result the exact convergence rate at zero of Lagrange interpolation polynomial to spaced nodes in [-1,1] due to M.Revers concerning f（x） = ｜x｜α with on equally     

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.     

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.     

Rs空间中的Lagrange插值

本文给出了构造空间π^s_n中Lagrange插值适定结点组的添加超平面法以及构造沿无重复分量代数超曲面插值适定结点组的添加超平面法,从而弄清楚了这两种适定结点组间的几何结构。     

THE DIVERGENCE OF LAGRANGE INTERPOLATION FOR ｜x｜^α（2〈α〈4）AT EQUIDISTANT NODES

It is a classical result of Bernstein that the sequence of Lagrange interpolation polumomials to ｜x｜ at equally spaced nodes in [-1, 1] diverges everywhere, except at zero and the end-points. In the present paper, toe prove that the sequence of Lagrange interpolation polynomials corresponding to ｜x｜^α （2 〈 α 〈 4） on equidistant nodes in [-1, 1] diverges everywhere, except at zero and the end-points.     

函数逼近中的Newton和Lagrange插值多项式

讨论了Newton及Lagrange插值多项式在函数逼近中的联系和区别.     

On the Zero-Divergence of Equidistant Lagrange Interpolation

 In 1942, P. Szász published the surprising result that if a function f is of bounded variation on [−1, 1] and continuous at 0 then the sequence of the equidistant Lagrange interpolation polynomials converges at 0 to . In the present note we give a construction of a function continuous on [−1, 1] whose Lagrange polynomials diverge at 0. Moreover, we show that the rate of divergence attains almost the maximal possible rate. (Received 2 February 2000)     

On the Zero-Divergence of Equidistant Lagrange Interpolation

 In 1942, P. Szász published the surprising result that if a function f is of bounded variation on [−1, 1] and continuous at 0 then the sequence of the equidistant Lagrange interpolation polynomials converges at 0 to . In the present note we give a construction of a function continuous on [−1, 1] whose Lagrange polynomials diverge at 0. Moreover, we show that the rate of divergence attains almost the maximal possible rate.     

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.     

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.     

Simultaneous Approximation of Sobolev Classes by Piecewise Cubic Hermite Interpolation

For the approximation in $L_p$-norm, we determine the weakly asymptotic orders for the simultaneous approximation errors of Sobolev classes by piecewise cubic Hermite interpolation with equidistant knots. For $p = 1$, $∞$, we obtain its values. By these results we know that for the Sobolev classes, the approximation errors by piecewise cubic Hermite interpolation are weakly equivalent to the corresponding infinite-dimensional Kolmogorov widths. At the same time, the approximation errors of derivatives are weakly equivalent to the corresponding infinite-dimensional Kolmogorov widths.     

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.     

Lebesgue constant for Lagrange interpolation on equidistant nodes

Properties of Lebesgue function for Lagrange interpolation on equidistant nodes are investigated. It is proved that Lebesgue function can be formulated both in terms of a hypergeometric function 2F1 and Jacobt polynomials. Moreover, an integral expression of Lebesgue function is also obtained and the asymptotic behavior of Lebesgue constant is studied.     

基于Chebyshev多项式零点的Lagrange插值多项式逼近的注记

本文通过一个例子说明了文献[3]中定理6.9的不完善之处,并建立了:若f∈C^r_[-1,1],则     

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.     

THE DIVERGENCE OF LAGRANGE INTERPOLATION FOR |x|α (2＜α＜4) AT EQUIDISTANT NODES

It is a classical result of Bernstein that the sequence of Lagrange interpolation polynomials to |x| at equally spaced nodes in [-1, 1] diverges everywhere, except at zero and the end-points. In the present paper, we prove that the sequence of Lagrange interpolation polynomials corresponding to |x|α(2 ＜α＜ 4) on equidistant nodes in [-1,1] diverges everywhere, except at zero and the end-points.