Mean Convergence of Extended Lagrange Interpolation for Exponential Weights

In this paper, we complete our investigations of mean convergence of Lagrange interpolation for fast decaying even and smooth exponential weights on the line. In doing so, we also present a summary of recent related work on the line and [–1,1] by the authors, Szabados, Vertesi, Lubinsky and Matjila. We also emphasize the important and fundamental ideas, applied in our proofs, that were developed by Erds, Turan, Askey, Freud, Nevai, Szabados, Vértesi and their students and collaborators. These methods include forward quadrature estimates, orthogonal expansions, Hilbert transforms, bounds on Lebesgue functions and the uniform boundedness principle.     

Mean Convergence of Derivatives of Extended Lagrange Interpolation with Additional Nodes

Weighted L^p convergence of derivatives of extended Lagrange interpolation at the union of zeros of generalized Jacobi polynomials and some additional points is investigated.     

On Quadrature Convergence of Extended Lagrange Interpolation

Quadrature convergence of the extended Lagrange interpolant for any continuous function is studied, where the interpolation nodes are the zeros of an orthogonal polynomial of degree and the zeros of the corresponding ``induced' orthogonal polynomial of degree . It is found that, unlike convergence in the mean, quadrature convergence does hold for all four Chebyshev weight functions. This is shown by establishing the positivity of the underlying quadrature rule, whose weights are obtained explicitly. Necessary and sufficient conditions for positivity are also obtained in cases where the nodes and interlace, and the conditions are checked numerically for the Jacobi weight function with parameters and . It is conjectured, in this case, that quadrature convergence holds for .

     

Convergence of Modified Lagrange Interpolation toLp-Functions Based on the Zeros of Orthonormal Polynomials with Freud Weights

We consider the “Freud weight”W²_Q(x)=exp(−Q(x)). let 1<p<∞, and letL*_n(f) be a modified Lagrange interpolation polynomial to a measurable functionf∈{f; ess sup_x∈ |f(x)| W_Q(x)(1+|x|)^α<∞},α>0. Then we have lim_n→∞ ∫^∞_−∞ [|f(x)−L*_n(f; x)| W_Q(x)(1+|x|)^−Δ]^p dx=0, whereΔis a constant depending onpandα.     

On Mean Convergence of Lagrange Interpolation for General Arrays

For n1, let {x_jn}ⁿ_j=1 be n distinct points in a compact set K and letL_n[·] denote the corresponding Lagrange interpolation operator. Let v be a suitably restricted function on K. What conditions on the array {x_jn}_1jn, n1 ensure the existence of p>0 such that lim_n→∞ (f−L_n[f]) v_Lp(K)=0 for very continuous f: K→ ? We show that it is necessary and sufficient that there exists r>0 with sup_n1 π_nv_Lr(K) ∑ⁿ_j=1 (1/|π′_n| (x_jn))<∞. Here for n1, π_n is a polynomial of degree n having {x_jn}ⁿ_j=1 as zeros. The necessity of this condition is due to Ying Guang Shi.     

On Weighted Mean Convergence of Lagrange Interpolation for General Arrays

For n1, let {x_jn}_j=1ⁿ be n distinct points and let L_n[·] denote the corresponding Lagrange Interpolation operator. Let W : →[0,∞). What conditions on the array {x_jn}_{1jn, n1} ensure the existence of p>0 such

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for every continuous f : → with suitably restricted growth, and some “weighting factor” φ^b? We obtain a necessary and sufficient condition for such a p to exist. The result is the weighted analogue of our earlier work for interpolation arrays contained in a compact set.     

Uniform or Mean Convergence of Hermite–Fejér Interpolation of Higher Order for Freud Weights

In this paper we show the uniform or mean convergence of Hermite–Fejér interpolation polynomials of higher order based at the zeros of orthonormal polynomials with the typical Freud weight.     

扩充的Hermite-Fejér插值算子平均收敛性

讨论了以Ｊａｃｏｂｉ正交多项式零点为插值结点的扩充Ｈｅｒｍｉｔｅ－Ｆｅｊｅｒ插值算子在Ｌｐｕ空间的平均收敛性。首先给出了算子加权平均收敛的条件,进一步得到了收敛阶。     

Mean Convergence of Lagrange Type Interpolation on an Arbitrary System of Nodes

Abstract Sufficient conditions of convergence and rate of convergence for Lagrange type interpolation in theWeighted L~p norm on an arbitrary system of nodes are given.     

基于LAGRANGE插值的高阶微分中值定理

本文基于LAGRANGE插值，将微积分中非常重要的中值定理推广到了高阶的情形。     

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.     

Convergence of Lagrange Interpolation Polynomials for Piecewise Smooth Functions

The present paper first establishes a decomposition result for f(x)∈ C ^r C ^r+1. By using this decomposition we thus can obtain an estimate of ∣f(x) - L _n (f,x)∣ which reflects the influence of the position of the x's and ω(f ^(r+1),δ)_j, j = 0,1,...,s, on the error of approximation. This revised version was published online in June 2006 with corrections to the Cover Date.     

扩充的Hermite—Fejer插值算子平均收敛性

讨论了以Ｊａｃｏｂｉ正交多项式零点为插值结点的扩充Ｈｅｒｍｉｔｅ－Ｆｅｊｅｒ插值算子在Ｌｕ＾ｐ空间的平均收敛性。首先给出了算子加权平均收敛的条件,进一步得到了收敛阶。     

Convergence of Gaussian Quadrature and Lagrange Interpolation in Haar Systems

Consider a Markov system of functions whose linear span is dense with respect to the uniform norm in the space of the continuous functions on a finite interval. Gaussian rules are those which correctly integrate as many successive basis functions as possible with the lesser number of nodes. In this paper we provide a simple proof of the fact that such rules converge for all bounded Riemann-Stieltjes integrable functions. The proposed proof is also valid for any sequence of quadrature rules with positive coefficients which converge for the basis functions. Taking the nodes of the Gaussian rules as nodes for Lagrange interpolation, we give a sufficient condition for the convergence in L ₂-norm of such processes for bounded Riemann-Stieltjes integrable functions.     

关于一类Hermite-Fejer插值算子的平均收敛

本文给出基于{xk}n+1 k=0的Hermite-Fejer插值算子平均收敛的一些新结论,这里xo=1,xn+1=-1,xk(k=1,2,…,n)是n阶Jacobi多项式的零点.     

Weighted Least Square Convergence of Lagrange Interpolation on the Unit Circle

In the paper, a result of Walsh and Sharma on least square convergence of Lagrange interpolation polynomials based on the n-th roots of unity is extended to Lagrange interpolation on the sets obtained by projecting vertically the zeros of (1-x)²=P ^(a,) _n(x),a>0,>0,(1-x)P^(a,) _n(x),a>0,>-1,(1+x)P P^(a,) _n(x),a>-1,0 and P^(a,) _n(x),a>-1,>-1, respectively, onto the unit circle, where P^(a,) _n(x),a>-1,>-1, stands for the n-th Jacobi polynomial. Moreover, a result of Saff and Walsh is also extended.     

Weighted Least Square Convergence of Lagrange Interpolation on the Unit Circle

In the paper, a result of Walsh and Sharma on least square convergence of Lagrange interpolation polynomials based on the n-th roots of unity is extended to Lagrange interpolation on the sets obtained by projecting vertically the zeros of (1-x)²=P ^(a,β) _n(x),a>0,β>0,(1-x)P^(a,β) _n(x),a>0,β>-1,(1+x)P P^(a,β) _n(x),a>-1,β0 and P^(a,β) _n(x),a>-1,β>-1, respectively, onto the unit circle, where P^(a,β) _n(x),a>-1,β>-1, stands for the n-th Jacobi polynomial. Moreover, a result of Saff and Walsh is also extended.     

On the Lebesgue Function of Weighted Lagrange Interpolation. I. (Freud-Type Weights)

For a wide class of Freud-type weights of form w = exp(-Q) we investigate the behavior of the corresponding weighted Lebesgue function λ _n (w,X,x) , where X = { x _kn } (-∞,∞) is an interpolatory matrix. We prove that for arbitrary X (-∞,∞) and ɛ > 0 , fixed, λ _n (w, X, x) ≥ c ɛ log n, x ∈ [-a _n , a _n ]\H _n , n ≥ 1, where a _n is the MRS number and |H _n | ≤ 2 ɛ a _n . The result corresponds to the behavior of the ``ordinary' Lebesgue function in [-1,1] . Other exponential weights are considered in our subsequent paper. October 28, 1996. Date revised: April 7, 1997. Date accepted: March 18, 1998.     

关于一类Hermite-Fejér插值算子的平均收敛

本文给出基于{xk}_(k=0)~(n+1)的Hermite-Fejér插值算子平均收敛的一些新结论,这里x0=1,xn+1=-1,xk（k=1,2,…,n）是n阶Jacobi多项式的零点.     

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.