渐近于Hermite多项式的双正交系统

本文利用渐近于Gauss函数的函数类?,给出渐近于Hermite正交多项式的一类Appell多项式的构造方法,使得该序列与?的n阶导数之间构成了一组双正交系统.利用此结果,本文得到多种正交多项式和组合多项式的渐近性质.特别地,由N阶B样条所生成的Appell多项式序列恰为N阶Bernoulli多项式.从而,Bernoulli多项式与B样条的导函数之间构成了一组双正交系统,且标准化之后的Bernoulli多项式的渐近形式为Hermite多项式.由二项分布所生成的Appell序列为Euler多项式,从而,Euler多项式与二项分布的导函数之间构成一组双正交系统,且标准化之后的Euler多项式渐近于Hermite多项式.本文给出Appell序列的生成函数满足的尺度方程的充要条件,给出渐近于Hermite多项式的函数列的判定定理.应用该定理,验证广义Buchholz多项式、广义Laguerre多项式和广义Ultraspherical(Gegenbauer)多项式渐近于Hermite多项式的性质,从而验证超几何多项式的Askey格式的成立.     

构造Hermite插值“基函数”的方法

给出了一种构造Hermite插值"基函数"的方法,画出了"基函数"的构造图.借助于这组"基函数"的线性组合来求Hermite插值多项式,计算过程非常简单.之后,把这种求"基函数"的方法推广到了二元Hermite插值中,为二元Hermite插值"基函数"的构造提供了一种简单实用的方法.     

与Hermite多项式和广义Laguerre多项式相关联的双正交多项式系统

生成函数刻画了正交多项式的很多重要性质.本文的主要目的是根据生成函数的特点研究正交多项式类之间的渐近关系.本文拓展了Lee及其合作者的工作,构造一类双正交多项式系统,并由此构造出分别渐近于Hermite多项式和广义Laguerre多项的函数列;给出渐近于Hermite多项式和广义Laguerre多项的函数列的判定定理.作为这些性质的应用,可以直接获得若干正交多项式和组合多项式的渐近表示,从而验证了揭示超几何多项式渐近关系的Askey格式成立.     

ECT插值余项的表示及其应用

在多项式插值理论及样条逼近中,Hermite插值多项式余项的讨论是很重要的。在[1,2]中,给出了一系列Hermite插值多项式余项的表达式,特别是各阶导数余项的表达式。还运用这些表达式讨论了样条函数,给出其余项估计和渐近展开。 随着样条理论的发展,已经用其它函数系代替多项式组成了各种样条函数空间,其中最引人注目的是ECT样条。Pruess讨论的张力样条及C.A.Micchelli讨论的?-样     

分段低次插值多项式序列的一致收敛性

利用分段线性与三次Hermite插值基函数以及连续模概念,分别推导出分段线性与三次Hermite插值多项式序列一致收敛于被插函数.     

论Hermite插值

本文给出了Hermite插值多项式及其各阶导数的显式表示. 对于一个在x的某个领域内有足够高阶连续导数的函数f和位于该领域的任意一组节点, 给出了用f的Hermite插值多项式在点x的任意阶导数逼近f(x)的相应导数时余项的渐近表示.     

Hermite四点插指公式

文章利用Hermite插值基函数,将求解Hermite四点插指问题转换为求解8个派生出来的多项式插值问题,证明了Hermite四点插指公式的存在唯一性,并用两种方法构造出Hermite四点插指公式,最后给出了一个算例.     

多项式插值余项定理的一个自然证明

通过对插值多项式函数性质进行分析,多项式插值余项的基本形式得到诱导,再从该基本形式出发,获得了多项式插值余项定理的新证明.整个证明过程无需借助辅助函数的构造,因而显得较为自然.这种自然证明的方式也可用于Hermite切触型插值多项式余项的证明.     

关于Hermite插值的注记

设 H_n(x)是在节点 x_0,x_1,…,x_n 上插值 f(x)的 n 次 Hermite 插值多项式.最近[1]用函数 f 的差商给出了 H_n(x) 的表达式.这里指出:这一表达式实际已有 (例如参见[2]),函数 f 的 n 次 Hermite 插值多项式 H_n(x) 及其余项可用 f 的差商简单地表示为     

Tchebycheff插值余项及其在T-样条中的应用

在多项式逼近理论及样条逼近的讨论中,Hermite多项式余项讨论是很重要的。作者在以前一系列工作中(〔1,2〕),对于插值Hermite多项式的余项给出一系列表达式,特别是各阶导数余项的表达式。运用这些表达式成功地讨论了一系列样条函数。给出它们的余项估计和渐近展开。     

Best one-sided L1 approximation to the Heaviside and sign functions

We find the polynomials of the best one-sided approximation to the Heaviside and sign functions. The polynomials are obtained by Hermite interpolation at the zeros of some Jacobi polynomials. Also we give an estimate of the error of approximation and characterize the extremal points of the convex set of the best approximants.     

On the solvability of a system of forward-backward linear equations with unbounded operator coefficients

An estimate of the norm of the Lagrange interpolation operator in the multidimensional weighted Sobolev space is obtained. It is shown that, under a certain choice of the sequence of multi-indices, the interpolating polynomials converge to the interpolated function and the rate of convergence is of the order of the best approximation of this function by algebraic polynomials in this space.     

Approximation of trigonometric functions by trigonometric polynomials with interpolation

The paper studies the approximation order of periodic functions by trigonometric polynomials with interpolation in arbitrary set of nodes. A method of construction of Hermite interpolation polynomials is pointed out.     

On the problem of Hermite interpolation of operators in a Hilbert space

The problem of Hermite operator interpolation with interpolational conditions containing Gateaux higher-order differentials in arbitrary directions is investigated. A necessary and sufficient condition for solvability of this problem in a Hilbert space is established, and the set of all Hermite operator polynomials and its subset of interpolants preserving operator polynomials of the same degree are described. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 78, 1994, pp. 38–48.     

Hermite–Fejér interpolation operator and characterization of functions

In this paper we investigate the approximation behaviour of the so‐called Hermite–Fejér interpolation operator based on the zeros of Jacobi polynomials. As a result we obtain the asymptotic formula of approximation rate for these operators. Moreover, such a formula is valid for any individual continuous function. We will also study the K ‐functional deduced by this operator. Consequently the asymptotic term of this K ‐functional is established. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computable Error Estimates for the Collocation Method Applied to Two-point Boundary Value Problems

A two-point boundary value problem is solved with the collocationmethod using piecewise Hermite polynomials. Polynomials of thesame form are then used to interpolate the analytical solutionof the differential equation. Using this interpolation error,an estimate of the error in the collocation approximation iscalculated. The behaviour of the method is illustrated by linearand non-linear numerical examples.     

Estimate of the norm of the Lagrange interpolation operator in a multidimensional Sobolev space

We obtain an estimate of the norm of the Lagrange interpolation operator in a multidimensional Sobolev space. It is shown that, under a suitable choice of the sequence of multi-indices, interpolation polynomials converge to the interpolated function and their rate of convergence is of the order of the best approximation of this function.     

Error estimates for Hermite interpolation on spheres

In this paper, we prove convergence rates for spherical spline Hermite interpolation on the sphere S^d−1 via an error estimate given in a technical report by Luo and Levesley. The functionals in the Hermite interpolation are either point evaluations of pseudodifferential operators or rotational differential operators, the desirable feature of these operators being that they map polynomials to polynomials. Convergence rates for certain derivatives are given in terms of maximum point separation.     

On the pointwise estimations of approximation of functions and their derivatives by Hermite interpolation

The object of this paper is to establish the pointwise estimations of approximation of functions in C¹ and their derivatives by Hermite interpolation polynomials. The given orders have been proved to be exact in general.     

A constructive approach to nodal splines

In the context of local spline interpolation methods, nodal splines have been introduced as possible fundamental functions by de Villiers and Rohwer in 1988. The corresponding local spline interpolation operator possesses the desirable property of reproducing a large class of polynomials. However, it was remarked that their definition is rather intricate so that it seems desirable to reveal the actual origin of these splines. The real source can be found in the Martensenoperator which can be obtained by two-point Hermite spline interpolation problem posed and proved by Martensen [Darstellung und Entwicklung des Restgliedes der Gregoryschen Quadraturformel mit Hilfe von Spline-Funktionen, Numer. Math. 21(1973)70–80]. On the one hand, we will show how to represent the Hermite Martensen spline recursively and, on the other hand, explicitly in terms of the B-spline by using the famous Marsden identity. Having introduced the Martensenoperator, we will show that the nodal spline interpolation operator can be obtained by a special discretization of the occurring derivatives. We will consider symmetric nodal splines of odd degree that can be obtained by our methods in a natural way.