<Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> Hermite interpolation by Pythagorean hodograph quintics in Minkowski space

Curves in the Minkowski space are very well suited to describe the medial axis transform (MAT) of planar domains. Among them, Minkowski Pythagorean hodograph (MPH) curves correspond to domains where both the boundaries and their offsets admit rational parameterizations (Choi et al., Comput Aided Design 31:59–72, 1999; Moon, Comput Aided Geom Design 16:739–753; 1999). We construct MPH quintics which interpolate two points with associated first derivative vectors and analyze the properties of the system of solutions, including the approximation order of the ‘best’ interpolant.      

Jackson type theorems on complex curves

Let Γ be a closed smooth Jordan curve in the complex plane. In this paper, with the help of a class of fundamental functions of Hermite interpolation, the author introduces a continuous function interpolation which uniformly approximates to f(z) ε C(Γ) with the same order of approximation as that in Jackson Theorem 1 on real interval [−1, 1]. The accuracy of the order of approximation is proved. Using the method different from the early works, the author studies simultaneous approximation to function and its derivatives and the desired results analogues to that in Jackson Theorem 2 on real interval [−1, 1] are obtained.      

Arc Length Estimation and the Convergence of Polynomial Curve Interpolation

When fitting parametric polynomial curves to sequences of points or derivatives we have to choose suitable parameter values at the interpolation points. This paper investigates the effect of the parameterization on the approximation order of the interpolation. We show that chord length parameter values yield full approximation order when the polynomial degree is at most three. We obtain full approximation order for arbitrary degree by developing an algorithm which generates more and more accurate approximations to arc length: the lengths of the segments of an interpolant of one degree provide parameter intervals for interpolants of degree two higher. The algorithm can also be used to estimate the length of a curve and its arc-length derivatives. AMS subject classification (2000) 65D05, 65D10     

高阶Hermite-Fejér型内插的一个性质

本文给出高阶 Herm ite- Fejér型内插的一个性质     

Hermite插值算子在Orlicz空间内的逼近

插值算子逼近是逼近论中一个非常有趣的问题,尤其是以一些特殊的点为结点的插值算子的逼近问题很受人们的关注.研究了以第一类Chebyshev多项式零点为插值结点的Hermite插值算子在Orlicz范数下的逼近.     

Martensen Splines

This paper is concerned with the construction of the fundamental functions associated with a two-point Hermite spline interpolation scheme used by Martensen in the context of the remainder of the Gregory quadrature rule. We derive both a recursive construction and an explicit representation in terms of the underlying B-Splines which can easily be deduced using Marsden’s identity. We can make use of these functions in order to introduce a local interpolation scheme which reproduces all splines. Finally, we examine the error of this interpolant to a sufficiently smooth function and realize that it behaves like in the case of splines of degree n. AMS subject classification (2000) 65D05, 65D07, 41A15     

Error estimates of Hermite interpolation

This paper presents a procedure for obtaining error estimates for Hermite interpolation at the Chebyshev nodes {cos ((2j+1)/2n)} _j ^=0n–1 –1x1, for functionsf(x) of various orders of continuity. The procedure is applicable in many cases when the usual Lagrangian error bound is not, and is a better bound, in general, when both are applicable.     

一类收敛的线性Hermite重心权有理插值

众所周知, Hermite有理插值比Hermite多项式插值具有更好的逼近性, 特别是对于插值点序列较大时, 但很难解决收敛性问题和控制实极点的出现. 本文建立了一类线性Hermite重心有理插值函数$r(x)$,并证明其具有以下优良性质: 第一, 在实数范围内无极点; 第二, 当$k=0,1,2$时,无论插值节点如何分布, 函数$r^{(k)}(x)$具有$O(h^{3d+3-k})$的收敛速度; 第三, 插值函数$r(x)$仅仅线性依赖于插值数据.     

Some convergence results for the generalized Padé-type approximants

The aim of this paper is to give some convergence results for some sequences of generalized Padé-type approximants. We will consider two types of interpolatory functionals: one corresponding to Langrange and Hermite interpolation and the other corresponding to orthogonal expansions. For these two cases we will give sufficient conditions on the generating functionG(x, t) and on the linear functionalc in order to obtain the convergence of the corresponding sequence of generalized Padé-type approximants. Some examples are given.     

A unified Pythagorean hodograph approach to the medial axis transform and offset approximation

Algorithms based on Pythagorean hodographs (PH) in the Euclidean plane and in Minkowski space share common goals, the main one being rationality of offsets of planar domains. However, only separate interpolation techniques based on these curves can be found in the literature. It was recently revealed that rational PH curves in the Euclidean plane and in Minkowski space are very closely related. In this paper, we continue the discussion of the interplay between spatial MPH curves and their associated planar PH curves from the point of view of Hermite interpolation. On the basis of this approach we design a new, simple interpolation algorithm. The main advantage of the unifying method presented lies in the fact that it uses, after only some simple additional computations, an arbitrary algorithm for interpolation using planar PH curves also for interpolation using spatial MPH curves. We present the functionality of our method for G¹ Hermite data; however, one could also obtain higher order algorithms.     

一类带参数的有理三次三角Hermite插值样条

给出一种带有参数的有理三次三角Hermite插值样条,具有标准三次Hermite插值样条相似的性质.利用参数的不同取值不但可以调控插值曲线的形状,而且比标准三次Hermite插值样条更好地逼近被插曲线.此外,选择合适的控制点,该种插值样条可以精确表示星形线和四叶玫瑰线等超越曲线.     

The Simultaneous Approximation Order by Hermite Interpolation in a Smooth Domain

Let D be a smooth domain in the complex plane. In D consider the simultaneous approximation to a function and its ith (0 ≤i≤q) derivatives by Hermite interpolation. The orders of uniform approximation and approximation in the mean, are obtained under some domain boundary conditions. Some known results are included as particular cases of the theorems of this paper. Received May 25, 2000, Revised November 3, 2000, Accepted December 7, 2000     

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 ) .     

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.     

ON THE SATURATION OF L_w~p-APPROXIMATION BY (O-q''''-q) TYPE HERMITE-FEJER INTERPOLATING POLYNOMIALS

The "o" saturation theorem and the degree of Lwp, approximation by (0 - q' - q) type Hermite-Fejer interpolating polynomials for mean convergence are obtained.