Über optimale Quadraturformeln mit freien Knoten zu Hilberträumen periodischer Funktionen

Summary For some special Hilbert-spaces of periodic analytic functions it is known that quadrature formulae of minimal norm with preassigned equidistant nodes are even so-called Wilf-formulae, i.e. they satisfy necessary conditions for minimal norm with respect to their nodes. By simple examples, however, it can be shown that equidistant Wilf-formulae are not necessarily optimal. In this paper the question of optimality of equidistant nodes in quadrature formulae for rather general Hilbert-spaces of periodic analytic functions is answered by giving sufficient conditions which can be interpreted as conditions on the size of the regularity-regions of the functions belonging to the Hilbert-spaces under consideration. Examples prove these conditions to be quite sharp.In addition the trapezoidal-rule is shown to be only optimal formula (with respect to the nodes and coefficients) of orderk.Finally the trapezoidal-rule is shown to be asymptotically optimal for wide classes of Hilbert-spaces of periodic functions.

     

基于等距节点积分公式的牛顿迭代法及其收敛阶

利用等距节点的数值积分公式构造牛顿迭代法的变形格式.我们证明了利用4等分5个节点的Newton-Cotes公式构造的变形牛顿迭代法收敛阶为3,并进一步证明了对于最常用的3等分4节点、5等分6节点、6等分7节点、7等分8节点积分公式,所得到的变形牛顿迭代法收敛阶都是3.最后,本文猜想,利用任意等分的积分公式构造变形牛顿迭代法,所得的迭代格式收敛阶都是3.     

一类Birkhoff型2-周期三角和仿三角插值(英文)

本文讨论了2π周期和反周期函数在等距结点上的一类Birkhoff型2-周期三角和仿三角插值问题,给出了此问题有解的充要条件,并构造出插值基。     

在等距节点处的反周期函数的(0,m_1,m_2,…,m_p)三角插值

研究了以π为周期的反周期函数的Birkhoff三角插值,解决了在等距节点处的反周期函数的(0,m1,m2,…,mp)三角插值问题,得到了解存在的条件.     

在等距节点处的反周期函数的（0，m1，m2，…，mp）三角插值

研究了以π为周期的反周期函数的Birkhoff三角插值,解决了在等距节点处的反周期函数的（0,m1,m2,…,mp）三角插值问题,得到了解存在的条件．     

Birkhoff型三角插值

1 引言 Birkhoff三角插值是近年来比较活跃的一个研究课题,涉及Birkhoff三角插值的研究文献也很多(如G.G.Lorentz~([1]),沈燮昌~([2])等综合性文章).     

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.     

Stability of quadrature rule methods for first kind volterra integral equations

Gladwin [4] proved that Newton-Gregory formulas of order larger than 2 produce unstable algorithms when applied to nonlinear Volterra integral equations of the first kind. It is shown that similar results are true for all interpolatory quadrature rules using equidistant nodes. Upper bounds for the error order of quadrature rules, which lead to stable methods are given. Some higher order stable methods are indicated.     

Local Approximation by Parabolic Splines in the Mean for Large Averaging Intervals

Mathematical Notes - Local parabolic splines on the axis $$\mathbb R$$ with equidistant nodes realizing the simplest local approximation scheme are considered. But, instead of the values of...     

On the Lebesgue constant of barycentric rational interpolation at equidistant nodes

Recent results reveal that the family of barycentric rational interpolants introduced by Floater and Hormann is very well-suited for the approximation of functions as well as their derivatives, integrals and primitives. Especially in the case of equidistant interpolation nodes, these infinitely smooth interpolants offer a much better choice than their polynomial analogue. A natural and important question concerns the condition of this rational approximation method. In this paper we extend a recent study of the Lebesgue function and constant associated with Berrut’s rational interpolant at equidistant nodes to the family of Floater–Hormann interpolants, which includes the former as a special case.     

一个双周期整(0,Δ_h~m)插值问题

本文研究等距结点上双周期整（０,Δｍｈ ）插值问题,得到它在 Ｂ２σ中有唯一解的充要条件,给出了这种插值函数的精确表达式,同时也考虑了该插值算子的收敛性     

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.     

Hermite spline interpolation in the discrete periodic case

Interpolation of discrete periodic complex-valued functions by the values and increments given at equidistant nodes is examined. A space of discrete functions in which the interpolation problem is uniquely solvable is introduced. Extremal and limit properties of the solution to this problem are found.     

ϕ-Strong approximation of functions by trigonometric polynomials

Mathematical Notes - The rate of ?-strong approximation of periodic functions by trigonometric polynomials constructed on the basis of interpolating polynomials with equidistant nodes is...     

Optimale Approximation mit Nebenbedingungen an lineare Funktionale auf periodischen Funktionen

Summary In a general Hilbert space of periodic functions numerical approximations with equidistant nodes for any bounded linear functional are given which are of minimal error norm in the class of approximations being exact for certain trigonometric polynomials. In examples optimal quadrature formulas with such side conditions are considered.     

Optimal interval length for the collocation of the Newton interpolation basis

Numerical Algorithms - It is known that the Lagrange interpolation problem at equidistant nodes is ill-conditioned. We explore the influence of the interval length in the computation of divided...     

Lower bound for the Lebesgue function of an interpolation process with algebraic polynomials on equidistant nodes of a simplex

For an interpolation process with algebraic polynomials of degree n on equidistant nodes of an m-simplex for m ≥ 2, we obtain a pointwise lower bound for the Lebesgue function similar to the well-known estimate for interpolation on a closed interval.     

On the Lebesgue constant of Berrut’s rational interpolant at equidistant nodes

It is well known that polynomial interpolation at equidistant nodes can give bad approximation results and that rational interpolation is a promising alternative in this setting. In this paper we confirm this observation by proving that the Lebesgue constant of Berrut’s rational interpolant grows only logarithmically in the number of interpolation nodes. Moreover, the numerical results suggest that the Lebesgue constant behaves similarly for interpolation at Chebyshev as well as logarithmically distributed nodes.     

On the Lebesgue constant of Berrut’s rational interpolant at equidistant nodes

It is well known that polynomial interpolation at equidistant nodes can give bad approximation results and that rational interpolation is a promising alternative in this setting. In this paper we confirm this observation by proving that the Lebesgue constant of Berrut’s rational interpolant grows only logarithmically in the number of interpolation nodes. Moreover, the numerical results suggest that the Lebesgue constant behaves similarly for interpolation at Chebyshev as well as logarithmically distributed nodes.     

On a generalization of compound Newton-Cotes quadrature formulas

We consider quadrature formulas defined by piecewise polynomial interpolation at equidistant nodes, admitting the nodes of adjacent polynomials to overlap, which generalizes the interpolation scheme of the compound Newton-Cotes quadrature formulas. The error constantse _,n in the estimate