On the poisedness of Bojanov–Xu interpolation

In this paper, we consider the bivariate Hermite interpolation introduced by Bojanov and Xu [SIAM J. Numer. Anal. 39(5) (2002) 1780–1793]. The nodes of the interpolation with Π_2k-δ, where δ=0 or 1, are the intersection points of 2k+1 distinct rays from the origin with a multiset of k+1-δ concentric circles. Parameters are the values and successive radial derivatives, whenever the corresponding circle is multiple. The poisedness of this interpolation was proved only for the set of equidistant rays [Bojanov and Xu, 2002] and its counterparts with other conic sections [Hakopian and Ismail, East J. Approx. 9 (2003) 251–267]. We show that the poisedness of this (k+1-δ)(2k+1) dimensional Hermite interpolation problem is equivalent to the poisedness of certain 2k+1 dimensional Lagrange interpolation problems. Then the poisedness of Bojanov–Xu interpolation for a wide family of sets of rays satisfying some simple conditions is established. Our results hold also with above circles replaced by ellipses, hyperbolas, and pairs of parallel lines.Next a conjecture [Hakopian and Ismail, J. Approx. Theory 116 (2002) 76–99] concerning a poisedness relation between the Bojanov–Xu interpolation, with set of rays symmetric about x-axis, and certain univariate lacunary interpolations is established. At the end the poisedness for a wide class of lacunary interpolations is obtained.     

Polynomial Interpolation on the Unit Sphere and on the Unit Ball

The problem of interpolation on the unit sphere S ^d by spherical polynomials of degree at most n is shown to be related to the interpolation on the unit ball B ^d by polynomials of degree n. As a consequence several explicit sets of points on S ^d are given for which the interpolation by spherical polynomials has a unique solution. We also discuss interpolation on the unit disc of R ² for which points are located on the circles and each circle has an even number of points. The problem is shown to be related to interpolation on the triangle in a natural way.     

Polynomial interpolation on the unit sphere II

The problem of interpolation at (n+1)² points on the unit sphere by spherical polynomials of degree at most n is proved to have a unique solution for several sets of points. The points are located on a number of circles on the sphere with even number of points on each circle. The proof is based on a method of factorization of polynomials. Dedicated to Mariano Gasca on the occasion of his 60th birthday The second author was supported by the Graduate Program Applied Algorithmic Mathematics of the Munich University of Technology. The work of the third author was supported in part by the National Science Foundation under Grant DMS-0201669.     

Bivariate mean value interpolation on circles

In the paper we prove that the bivariate mean-value interpolation problem, where part of the interpolation parameters are integrals over concentric circles, is not poised.     

Harmonic interpolation of Hermite type based on Radon projections with constant distances

We study harmonic interpolation of Hermite type of harmonic functions based on Radon projections with constant distances of chords. We show that the interpolation polynomials are continuous with respect to the angles and the distances. When the chords coalesce to some points on the unit circle, we prove that the interpolation polynomials tend to a Hermite interpolation polynomial at the coalescing points.     

Lagrange插值和Hermite-Fejér插值在Wiener空间下的平均误差

在L_q-范数逼近的意义下,确定了基于Chebyshev多项式零点的Lagrange插值多项式列和Hermite-Fejér插值多项式列在Wiener空间下的p-平均误差的弱渐近阶.从我们的结果可以看出,当2≤q<∞,1≤p<∞时,基于第一类Chebyshev多项式零点的Lagrange插值多项式列和Hermite-Fejér插值多项式列的p-平均误差弱等价于相应的最佳逼近多项式列的p-平均误差.在信息基计算复杂性的意义下,如果可允许信息泛函为计算函数在固定点的值,那么当1≤p,q<∞时,基于第一类Chebyshev多项式零点的Lagrange插值多项式列和Hermite-Fejér插值多项式列在Wiener空间下的p-平均误差弱等价于相应的最小非自适应p-平均信息半径.     

Fitting concentric circles to measurements

Measurements for fitting a given number of concentric circles are recorded. For each concentric circle several measurements are taken. The problem is to fit the given number of circles to the data such that all circles have a common center. This is a generalization of the problem of fitting a set of points to one circle. Three objectives, to be minimized, are considered: the least squares of distances from the circles, the maximum distance from the circles, and the sum of the distances from the circles. Very efficient optimal solution procedures are constructed. Problems based on a total of 10,000 measurements are solved in about 10 s with the least squares objective, $<$ 2 s with the maximum distance objective, and a little more than 1 min for the minisum objective.     

General Interpolation Formulae for Barycentric Blending Interpolation

General interpolation formulae for barycentric interpolation and barycentric rational Hermite interpolation are established by introducing multiple parameters,which include many kinds of barycentric interpolation and barycentric rational Hermite interpolation. We discussed the interpolation theorem, dual interpolation and special cases. Numerical example is given to show the effectiveness of the method.     

有理插值中不可达点的研究

通过对一元Thiele型连分式插值和二元Newton-Thiele型混合有理插值中不可达点的分析,给出了一种判断不可达点的方法.而且,对于任意给定的插值条件,通过构造带参数的Thiele型切触插值和二元Newton-Thiele型混合切触有理插值,使得不可达点变成可达点.数值例子也说明了这种方法的有效性.     

一类Hermite分形插值函数的构造算法及其运算性质

已知结点处的函数值和一阶导数值,给出了构造一类二次分形插值函数的方法.不同于仿射分形插值函数,得到的插值函数具有可微性,并讨论分形插值函数的微积分运算,最后给出一个构造例子.     

On multivariate Hermite interpolation

We study the problem of Hermite interpolation by polynomials in several variables. A very general definition of Hermite interpolation is adopted which consists of interpolation of consecutive chains of directional derivatives. We discuss the structure and some aspects of poisedness of the Hermite interpolation problem; using the notion of blockwise structure which we introduced in [10], we establish an interpolation formula analogous to that of Newton in one variable and use it to derive an integral remainder formula for a regular Hermite interpolation problem. For Hermite interpolation of degreen of a functionf, the remainder formula is a sum of integrals of certain (n + 1)st directional derivatives off multiplied by simplex spline functions.     

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.     

截面数据插值的组合模型及其实证

采用Moran指标分析截面数据空间自相关性.针对具有空间自相关截面数据缺失插值问题,分别建立一阶空间自回归插值模型与克立格方法插值模型,在此基础上建立截面数据的组合插值模型,并用信息熵法确定组合插值模型的加权系数.用2003年福建部分市县城镇化水平的截面数据建立插值模型实证研究结果表明:组合插值模型的效果优于单项插值模型的效果.     

球面上的曲面插值

<正>1引言随着现代工业生产的飞速发展,航空、气象、环境监测等领域需要研究解决限制在曲面上的四维数据插值问题,即由有限个位置处的信息推测其它若干位置点的信息.例如,地球上某个地区的温度分布、降雨量分布、大气层的"温室效应"等;飞行器(飞机、火箭、导弹等)表面压力分布规律、肿瘤的生长规律等.这些在数学上都可归结为限制在曲面上的曲面插值与逼近问题.这个问题自Barnhill提出以后,人们针对限制在球面上     

基于面积约束的统计直方图曲线拟合

我们提出用分段三次Hermite插值曲线拟合统计直方图的新方法.先根据统计直方图的特点选取Hermite插值曲线在插值点处的导数值和可调整的插值点,然后根据面积约束确定调整值,从而得到拟合曲线.所得拟合曲线与统计直方图有面积相等的约束,并且拟合曲线是C1连续的光滑曲线.所给方法简单、实用.     

Multigrid technique for biharmonic interpolation with application to dual and multiple reciprocity method

A biharmonic-type interpolation method is presented to solve 2D and 3D scattered data interpolation problems. Unlike the methods based on radial basis functions, which produce a large linear system of equations with fully populated and often non-selfadjoint and ill-conditioned matrix, the presented method converts the interpolation problem to the solution of the biharmonic equation supplied with some non-usual boundary conditions at the interpolation points. To solve the biharmonic equation, fast multigrid techniques can be applied which are based on a non-uniform, non-equidistant but Cartesian grid generated by the quadtree/octtree algorithm. The biharmonic interpolation technique is applied to the multiple and dual reciprocity method of the BEM to convert domain integrals to the boundary. This makes it possible to significantly reduce the computational cost of the evaluation of the appearing domain integrals as well as the memory requirement of the procedure. The resulting method can be considered as a special grid-free technique, since it requires no domain discretisation. This revised version was published online in June 2006 with corrections to the Cover Date.     

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.     

关于Newton—Thiele型二元有理插值的存在性问题

基于均差的牛顿插值多项式可以递归地实现对待插值函数的多项式逼近,而Thiele型插值连分式可以构造给定节点上的有理函数。将两者结合可以得到Newton-Thiele型二元有理插值（NTRI）算法,本文解决了NTRI算法的存在性问题,并有数值例子加以说明。     

参数曲线近似弧长参数化的三角函数方法

本描述了一种局部的近似弧长参数化插值方法，用三角函数对曲线的弧长函数进行分段逼近，段与段之间是相互独立的，且插值曲线在插值点处的弧长与原参数曲线的真实弧长相等。     

On the construction of interpolation mesh surfaces

A method for constructing two-dimensional interpolation mesh functions is proposed that is more flexible than the classical cubic spline method because it makes it possible to construct interpolation surfaces that fit the given function at specified points by varying certain parameters. The method is relatively simple and is well suited for practical implementation.