On the Uniform Convergence of Kergin Interpolation on the Disk

In this paper, we discuss the uniform convergences of Kergin interpolation and its derivatives for smooth functions defined on the disk. We also consider its application to the classical univariate interpolation.     

Kergin approximation in Banach spaces

We explore the convergence of Kergin interpolation polynomials of holomorphic functions in Banach spaces, which need not be of bounded type. We also investigate a case where the Kergin series diverges.     

On Spline Approximation for Singular Integral Equations on an Interval

This paper analyses the convergence of spline approximation methods for strongly elliptic singular integral equations on a finite interval. We consider collocation by smooth polynomial splines of odd degree multiplied by a weight function and a Galerkin-Petrov method with spline trial functions of even degree and piecewise constant test functions. We prove the stability of the methods in weighted Sobolev spaces and obtain the optimal orders of convergence in the case of graded meshes.     

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.     

Pointwise convergence of derivatives of Lagrange interpolation polynomials for exponential weights

For a general class of exponential weights on the line and on (−1,1), we study pointwise convergence of the derivatives of Lagrange interpolation. Our weights include even weights of smooth polynomial decay near ±∞ (Freud weights), even weights of faster than smooth polynomial decay near ±∞ (Erdös weights) and even weights which vanish strongly near ±1, for example Pollaczek type weights.     

一类插值多项式的导数逼近(英文)

本文给出了以雅可比多项式的零点作为插值节点的一类插值多项式 Bn( f ;x)的导数逼近具有一阶连续导数的函数的收敛阶 .并且指出 limn→∞ Bn′( f;-1 )≠f′( -1 ) .     

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.     

一类有理插值曲面模型及其可视化约束控制

本文构造一类新的基于函数值和偏导数值的双变量加权混合有理插值样条.与已有的有理插值样条相比,这类新的有理插值样条具有以下四方面的特性,其一,插值函数可以由简单的对称基函数来表示;其二,对任何正参数,插值函数满足C1连续,而且,在不限制参数取值的条件之下,插值曲面保持光滑;其三,插值函数不但含有参数,而且带有加权系数,增加了插值函数的自由度;其四,插值曲面的形状随着参数与加权系数的变化而变化.同时,本文讨论此类插值曲面的性质,包括基函数的性质、积分加权系数的性质和插值函数的边界性质.此类插值函数的优势在于,不改变给定插值数据的前提下,通过选择合适的参数和不同的加权系数,对插值区域内的任意点的函数值进行修改.因此可将其应用于曲面设计,根据实际设计需要,自由地修改曲面形状.数值实验表明,此类新的有理样条插值具有良好的约束控制性质.     

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.     

Smooth and Semismooth Newton Methods for Constrained Approximation and Estimation

In the article, we show that the constrained L ₂ approximation problem, the positive polynomial interpolation, and the density estimation problems can all be reformulated as a system of smooth or semismooth equations by using Lagrange duality theory. The obtained equations contain integral functions of the same form. The differentiability or (strong) semismoothness of the integral functions and the Hölder continuity of the Jacobian of the integral function were investigated. Then a globalized Newton-type method for solving these problems was introduced. Global convergence and numerical tests for estimating probability density functions with wavelet basis were also given. The research in this article not only strengthened the theoretical results in literatures but also provided a possibility for solving the probability density function estimation problem by Newton-type method.     

An Improvement to the Computing of Nonlinear Equation Solutions

In this paper, we suggest an improvement to the iterative methods based on the inverse interpolation polynomial, also referred to as the generalized Hermite interpolation, which increases the local order of convergence. A symbolic computation allows us to find the best coefficients with regard to the order of convergence. The adaptation of the strategy presented here gives a new iteration function with a new evaluation of the function. It also shows a smaller cost if we use adaptive multi-precision arithmetic. The numerical results computed using this system, with a floating point system representing 200 and 1000 decimal digits support this theory.     

非光滑函数的分数阶插值公式

本文基于局部分数阶Taylor展开式构造非光滑函数的分数阶插值公式,证明了插值公式的存在和唯一性,给出了分数阶插值的Lagrange表示形式及其误差余项,讨论了一种混合型的分段分数阶插值和整数阶插值的收敛阶.数值算例验证了对于非光滑函数分数阶插值明显优于通常的多项式插值,并说明在实际计算中采用分段混合分数阶和整数阶插值可以使得插值误差在区间上分布均匀,能够极大地提高插值精度.     

Gaussian rules on unbounded intervals

A quadrature rule as simple as the classical Gauss formula, with a lower computational cost and having the same convergence order of best weighted polynomial approximation in L¹ is constructed to approximate integrals on unbounded intervals. An analogous problem is discussed in the case of Lagrange interpolation in weighted L² norm. The order of convergence in our results is the best in the literature for the considered classes of functions.     

Spectral methods for weakly singular Volterra integral equations with pantograph delays

In this paper, the convergence analysis of the Volterra integral equation of second kind with weakly singular kernel and pantograph delays is provided. We use some function transformations and variable transformations to change the equation into a new Volterra integral equation with pantograph delays defined on the interval [-1, 1], so that the Jacobi orthogonal polynomial theory can be applied conveniently. We provide a rigorous error analysis for the proposed method in the L^∞-norm and the weighted L²-norm. Numerical examples are presented to complement the theoretical convergence results.     

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     

On two classes of approximation processes of integral type

The paper aims to study two classes of linear positive operators representing modifications of Picard and Gauss operators. The new operators reproduce both constants and a given exponential function. Approximation properties in polynomial weighted spaces are investigated and the speed of convergence is measured using a certain weighted modulus of smoothness. Also, the asymptotic behavior of the integral operators are established. Finally, aspects on generalized convexity are analyzed.

     

一类Hermite-Fejér插值算子的平均收敛(Ⅱ)

设X0=1,xn+1=-1,{xk}kn=1是n阶Jacobi多项式的零点,本文给出基于{xk)k=1 n+1 的Hermite-Fejer插值算子平均收敛的一些充要条件.     

A Jacobi-collocation method for solving second kind Fredholm integral equations with weakly singular kernels

In this work, we propose a Jacobi-collocation method to solve the second kind linear Fredholm integral equations with weakly singular kernels. Particularly, we consider the case when the underlying solutions are sufficiently smooth. In this case, the proposed method leads to a fully discrete linear system. We show that the fully discrete integral operator is stable in both infinite and weighted square norms. Furthermore, we establish that the approximate solution arrives at an optimal convergence order under the two norms. Finally, we give some numerical examples, which confirm the theoretical prediction of the exponential rate of convergence.     

基于负相协样本经验过程的加权弱收敛

在文中,我们建立了基于负相协样本经验过程的加权弱收敛,同时利用这个结果,我们还证明了其积分的弱收敛性,并将这些结果应用于生存分析,获得屯一些相应的弱极限性质。     

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

Abstract   Sufficient conditions of convergence and rate of convergence for Lagrange type interpolation in the weighted L ^p norm on an arbitrary system of nodes are given. Supported by the National Natural Sciences Foundation of China (No.19671082)