Lagrange插值在最大框架下的逼近误差

<正>1引言代数多项式插值理论是函数逼近理论和计算数学的重要研究内容.在函数逼近理论研究中,传统研究内容是对个体函数讨论插值多项式依赖于连续模或多项式最佳逼近的误差估计问题,其系列研究结果可见专著[7]或综述文章[8],近期研究结果可见[1,5]及     

基于Shifted Legendre多项式非线性年龄结构种群模型的数值解

基于Shifted Legendre多项式研究非线性年龄结构种群模型的数值解问题.定义了在区间[0,A]×[0,T]上函数的Shifted Legendre逼近多项式,通过Shifted Legendre算子矩阵结合Tau方法,把求解非线性年龄结构种群模型的数值解问题转化成非线性代数方程的求解问题.数值算例的结果显示该算法有效.     

基于结构元线性生成的复Fuzzy值函数项级数

文献[1]中提出了基于结构元理论的Fuzzy数项级数的概念,文献[2]、文献[3]、文献[4]对其收敛性进行了探讨,文献[5]、文献[6]对模糊值函数项数列及级数进行了研究。本文在此基础上给出了基于结构元线性生成的复Fuzzy值函数项数列及级数的定义,同时对复Fuzzy值函数项级数的一些重要性质进行了研究,并给出了相应定理。     

基于第二类Chebyshev多项式的双正交级数及共轭级数(英文)

记Un（z）是第二类Chebyshev多项式，伴随函数，这里讨论基于的双正交级数和其共轭级数的部分和逼近问题。     

H_q~p(01)空间中的多项式最佳逼近问题

文献[1]给出了H_q~p空间中函数的积分表达式,[2]、[3]、[4]在[1]的基础上研究了当P≥1时H_q~p空间的一些性质,[5]研究了当0相似文献   

关于具有拟单调系数的三角级数L1-收敛性的注记

假设三角级数的系数具有拟单调性，给出了级数按L^1[0，2π]中的范数收敛于其和函数的一个判别条件，推广了文献中的有关结果．     

V.D.变换的特性分析

函数f(x)的V.D.(Variation Diminishing)多项式样条逼近的若干特性已在[1—3]中作了一些讨论。本文从给定的一组型值点出发,给出了一类构造带有(或不带有)边界导数条件的V.D.多项式样条逼近的方法;并且证明了V.D.逼近具有“不产生多余拐点”(即与型值点二阶差符号变号个数相比较而言)这一优良的几何特性。此外,还给出了能保持型值点上二阶差符号的V.D.逼近方法。上述结果分别叙述在本文的§1—§3之中。本文的后一部分即§4、§5对V.D.逼近方法进行了一些误差分析。     

非周期神经网络及平移网络在L_w~p中的逼近

设s≥d≥1为整数, 1≤p≤+∞,借助于正交多元代数多项式系而构造了一类s维网络算子,并用于逼近Lpw[-1,1]s中的函数,给出了逼近的上界以及当此算子为平移网络算子及神经网络算子时的导数型估计.     

第二类 Hermite-Fejér多项式逼近度

近在[3]中验证了该多项式对这类函数的逼近效果也是很好的,它与最佳逼近多项式的逼近效果不相上下. 关于第二类eeb多项式零点作插值点时,稳定插值多项式(我们称其为第二类Hermite-Fejer多项式)的结果不多.最近见到Bojanic,Prasad和Saxena的结果,他们验证了第二类 Hermite-Fejer多项式(表达式的推导见[5]中的(1)):     

空间E_A与空间W~mE_A逼近问题的误差估计

§1.引言 在[1]中,用离散方法讨论了用梯形函数逼近E_A空间的元素,同时建立了用卷积作E_A空间元素的逼近,并且进一步研究了Orlicz-Sobolev空间的分段多项式逼近。本文的目的是建立[1]中所得结果的误差估计。为此,需要下面的记号: 设A(u)与B(v)是一对互补的N-函数,并记I=[0,1]。以L_A(I)表示满足     

Study on convergence of hybrid functions method for solution of nonlinear integral equations

In this article, we present the uniform convergence analysis and accuracy estimation of hybrid functions (HFs) method for finding the solution of nonlinear Volterra and Fredholm integral equations. The properties of HFs which consist of block-pulse functions (BPFs) and Legendre polynomials are used to reduce the solution of nonlinear integral equations to the solution of algebraic equations. The superiority and accuracy of the HFs method to BPF and Legendre polynomial methods are illustrated through some numerical examples.     

Application of a composition of generating functions for obtaining explicit formulas of polynomials

Using notions of composita and composition of generating functions, we obtain explicit formulas for the Chebyshev polynomials, the Legendre polynomials, the Gegenbauer polynomials, the Associated Laguerre polynomials, the Stirling polynomials, the Abel polynomials, the Bernoulli Polynomials of the Second Kind, the Generalized Bernoulli polynomials, the Euler Polynomials, the Peters polynomials, and the Narumi polynomials.     

Approximation properties of the Vallée-Poussin means of partial sums of a mixed series of Legendre polynomials

We estimate the order of weighted approximations of functions and their derivatives by using the means of mixed series of Legendre polynomials. As the main result, we obtain estimates of the order of approximation of a function and its derivatives by the Vallé-Poussin means and their derivatives.     

A hybrid approximation method for solving Hutchinson’s equation

The hybrid function approximation method for solving Hutchinson’s equation which is a nonlinear delay partial differential equation, is investigated. The properties of hybrid of block-pulse functions and Lagrange interpolating polynomials based on Legendre-Gauss-type points are presented and are utilized to replace the system of nonlinear delay differential equations resulting from the application of Legendre pseudospectral method, by a system of nonlinear algebraic equations. The validity and applicability of the proposed method are demonstrated through two illustrative examples on Hutchinson’s equation.     

Solution of nonlinear weakly singular Volterra integral equations using the fractional‐order Legendre functions and pseudospectral method

In this article, our main goal is to render an idea to convert a nonlinear weakly singular Volterra integral equation to a non‐singular one by new fractional‐order Legendre functions. The fractional‐order Legendre functions are generated by change of variable on well‐known shifted Legendre polynomials. We consider a general form of singular Volterra integral equation of the second kind. Then the fractional Legendre–Gauss–Lobatto quadratures formula eliminates the singularity of the kernel of the integral equation. Finally, the Legendre pseudospectral method reduces the solution of this problem to the solution of a system of algebraic equations. This method also can be utilized on fractional differential equations as well. The comparison of results of the presented method and other numerical solutions shows the efficiency and accuracy of this method. Also, the obtained maximum error between the results and exact solutions shows that using the present method leads to accurate results and fast convergence for solving nonlinear weakly singular Volterra integral equations. Copyright © 2015 John Wiley & Sons, Ltd.     

A new kind of inequality for Bessel functions

We study a new type of inequality for Bessel functions. This is an analog of an inequality for Legendre polynomials, which plays an important role in studying the nonlinear Boltzmann equation. As an application of the Bessel case we treat the spherical functions associated with Minkowski space.     

An effective approach for numerical solution of linear and nonlinear singular boundary value problems

In this study, an effective approach is presented to obtain a numerical solution of linear and nonlinear singular boundary value problems. The proposed method is constructed by combining reproducing kernel and Legendre polynomials. Legendre basis functions are used to get the kernel function, and then the approximate solution is obtained as a finite series sum. Comparison of numerical results is made with the results obtained by other methods available in the literature. Furthermore, efficiency and accuracy of the method are demonstrated in tabulated results and plotted graphs. The numerical outcomes demonstrate that our method is very effective, applicable, and convenient.     

On Legendre functions of imaginary degree and associated integral transforms

New integral representations, asymptotic formulas, and series expansions in powers of tanh(t/2) are obtained for the imaginary and real parts of the Legendre function P_iξ(cosht). Coefficients of these series expansions are orthogonal polynomials in the real variable ξ. A number of relations for these orthogonal polynomials are obtained on the basis of the generating function. Several inversion theorems are proven for the integral transforms involving the Legendre function of imaginary degree. In many cases it is preferable to employ these transforms, than Mehler-Fok transforms, since conditions placed on functions are less restrictive.     

A new operational vector approach for time-fractional subdiffusion equations of distributed order based on hybrid functions

This research study deals with the numerical solutions of linear and nonlinear time-fractional subdiffusion equations of distributed order. The main aim of our approach is based on the hybrid of block-pulse functions and shifted Legendre polynomials. We produce a novel and exact operational vector for the fractional Riemann–Liouville integral and use it via the Gauss–Legendre quadrature formula and collocation method. Consequently, we reduce the proposed equations to systems of equations. The convergence and error bounds for the new method are investigated. Six problems are tested to confirm the accuracy of the proposed approach. Comparisons between the obtained numerical results and other existing methods are provided. Numerical experiments illustrate the reliability, applicability, and efficiency of the proposed method.     

The expansion of the potential of a homogeneous ellipsoid in a series of tesseral harmonics

The Newtonian potential of a homogeneous triaxial ellipsoid is expanded in a series of tesseral harmonics (cf. Equation (1)) and the coefficients (cf. Equation (2)) are calculated (cf. Equation (25)). An integral formula of Legendre involving Legendre polynomials is generalized to associated Legendre functions (cf. Equation (16), (16^*)).