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.     

一种广义插值法

本文考虑一种广义插值问题，插值条件为小区间上的积分值，以弥补现有的插值方法在L2空间不再适用的不足，除了多项式插值外，还讨论了两种一次样条插值方法。     

A Kind of Generalization of the Curve Type Node Configuration in R^S（S 〉 2）

A kind of generalization of the Curve Type Node Configuration is given in this paper,and it is called the generalized node configuration CTNCB in RS(S＞2).The related multivariate polynomial interpolation problem is discussed.It is proved that the CTNCB is an appropriate node configuration for the polynomial space PSn (S＞2).And the expressions of the multivariate Vandermonde determinants that are related to the Odd Curve Type Node Configuration in R2 are also obtained.     

Lacunary Interpolation by Antiperiodic Trigonometric Polynomials

The problem of lacunary trigonometric interpolation is investigated. Does a trigonometric polynomial T exist which satisfies T(x _k) = a _k, D ^m T(x _k) = b _k, 0 k n – 1, where x _k = k/n is a nodal set, a _k and b _k are prescribed complex numbers, and m N. Results obtained by several authors for the periodic case are extended to the antiperiodic case. In particular solvability is established when n as well as m are even. In this case a periodic solution does not exist.     

General Structures of Block Based Interpolational Function

We construct general structures of one and two variable interpolation function,without depending on the existence of divided difference or inverse differences,and we also discuss the block based oscula...     

The real interpolation method on couples of intersections

Suppose that (X ₀, X ₁) is a Banach couple, X ₀ ∩ X ₁ is dense in X ₀ and X ₁, (X₀,X₁)_θq (0 < θ < 1, 1 ≤ q < ∞) are the spaces of the real interpolation method, ψ ∈ (X ₀ ∩ X ₁), ψ ≠ 0, is a linear functional, N = Ker ψ, and N _i stands for N with the norm inherited from X _i (i = 0, 1). The following theorem is proved: the norms of the spaces (N₀,N₁)_θ,q and (X₀,X₁)_θ,q are equivalent on N if and only if θ ? (0, α) ∪ (β_∞, α₀ ∪ (β₀, α_∞) ∪ (β, 1), where α, β, α₀, β₀, α_∞, and β _∞ are the dilation indices of the function k(t)=K(t,ψ;X ₀ ^* ,X ₁ ^* ).     

Multivariate polynomial interpolation under projectivities III: Remainder formulas

This is the third part of a note on multivariate interpolation. Some remainder formulas for interpolation on knot sets that are perspective images of standard lower data sets are given. They apply to all knot systems considered in parts I and II.Partially supported by PS900121.     

局部线性插值算子的一个构造方法

利用具有紧支集函数平移变换的拟任值是逼近论中的构造算子的一个重要方法,但拟插值算子一般不具有插值性质,本提供了一个简单的方法,该法可以构造出许多具有拟插值优点,又具有插值性质的线性算子。     

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.     

Bivariate Blending Thiele-Werner＇s Osculatory Rational Interpolation

Both the expansive Newton＇s interpolating polynomial and the Thiele-Werner＇s interpolation are used to construct a kind of bivariate blending Thiele-Werner＇s osculatory rational interpolation. A recursive algorithm and its characteristic properties are given. An error estimation is obtained and a numerical example is illustrated.     

一种关于函数值的二元有理插值方法

1 引言曲线曲面的构造和数学描述是计算机辅助几何设计中的核心问题．现在已有很多这种方法,如多项式样条方法、B-样条及非均匀B-样条(NURBS)方法、Bezier方法等等．这些方法已广泛应用于工业产品的形状设计,如飞机、轮船的外形设计．通常说来, 多项式样条方法一般都是插值型方法,插值曲线和插值曲面均通过插值点．构造这些多项式样条,其插值条件除插值点处的函数值外,一般还需要表示方向的导数值．但在很多实际问题中,导数值是很难得到的．同时,多项式样条方法的一个缺点是它的整体性质,在插值条件不变的情况下,在“插值函数关于插值条件的唯一性”的约束下,无法进行所构造的曲线曲面的整体或局部修改．NURBS方法和Bezier方法是所谓非插值型方法,用这些方法所构造出的曲线曲面一般不通过给定的点,给定的点是作为控制点出现的,通过给     

内插空间理论的应用

综述了线性算子内插法与内插空间理论在Banach空间几何学,微分算子,逼近理论,积分算子,Fourier分析等领域的一些应用。     

Bivariate Blending Thiele-Werner's Osculatory Rational Interpolation

Both the expansive Newton's interpolating polynomial and the Thiele-Werner's in- terpolation are used to construct a kind of bivariate blending Thiele-Werner's oscula- tory rational interpolation.A recursive algorithm and its characteristic properties are given.An error estimation is obtained and a numerical example is illustrated.     

CONVERGENCE OF (0,1,2,3) INTERPOLATION ON AN ARBITRARY SYSTEM OF NODES

1. IntroductionThis note deals with convergence of (0,1,2,3) illterpolation on an arbitrary system of nodes.Fisrt we illtroduce some definitions and notations.LetGiven a fiXed even integer m, let Ajk 6 Pm.--1 (the set of polynomials of degree at most mn-- 1)satisfyThen the (0,1,...,m--1) Hermite--Fej6r type illterpolation for f 6 C[--1, 1] is defined byand the (0,1,...,m--1) Hernilte interpolation for f e Cd--'[--1, 1] is defined by(of. [6]). We also need a well known fact:where 11' 11 sta…     

UNIFORM CONVERGENCE OF HERMITE INTERPOLATION OF HIGHER ORDER

In this paper the uniform convergence of Hermite-Fejer interpolation and Griinwald type theorem of higher order on an arbitrary system of nodes are presented.     

向量值切触有理插值存在性的判别方法

在用广义Vandermonde行列式给出Hermite插值多项式的表达式的基础上,针对a<,i>=2(i=1,2,…,s)的情形给出向量值切触有理插值存在性问题有解的条件及表达式.     

BLOCK BASED NEWTON-LIKE BLENDING OSCULATORY RATIONAL INTERPOLATION

With Newton＇s interpolating formula, we construct a kind of block based Newton-like blending osculatory interpolation.The interpolation provides us many flexible interpolation schemes for choices which include the expansive Newton＇s polynomial inter- polation as its special case. A bivariate analogy is also discussed and numerical examples are given to show the effectiveness of the interpolation.     

ON HIGH DIMENSIONAL INTERPOLATION

1　IntroductionIn this paper,we letPn,… ,nsn′s( or Psn) be the ( s-variate) polynomial space of all real ( s-variate) polynomials with the degreeof each variate atmostn and use the usual multivariate notationwj =wj11 … wjss,| j| =j1 +… + js( j1 ,… ,js∈ Z+) .[1 ] and [2 ] have discussed the Cross Type Node Configuration ( CRTNC) and thecorresponding bivariate interpolation in R2 .In this paper,we considerRs={ ( w1 ,… ,ws) :wi ∈ R,i =1 ,… ,s} .We say that s( s-1 ) -dimensional h…     

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.