A weighted bivariate blending rational interpolation based on function values

This paper presents a new weighted bivariate blending rational spline interpolation based on function values. This spline interpolation has the following advantages: firstly, it can modify the shape of the interpolating surface by changing the parameters under the condition that the values of the interpolating nodes are fixed; secondly, the interpolating function is C¹-continuous for any positive parameters; thirdly, the interpolating function has a simple and explicit mathematical representation; fourthly, the interpolating function only depends on the values of the function being interpolated, so the computation is simple. In addition, this paper discusses some properties of the interpolating function, such as the bases of the interpolating function, the matrix representation, the bounded property, the error between the interpolating function and the function being interpolated.     

一种基于函数值的二元有理插值函数及其性质

利用带参数的仅以被插函数的函数值作为插值条件的一元有理插值方法,构造了一种分母为双二次的仪基于函数值的二元有理双三次插值函数,插值函数具有简洁的显示表示,插值函数中含有四个参数,当这些参数满足一定条件时,插值曲而在插值区域上C1光滑.由于插值函数中含有参数,这样可以在插值数据不变的情况下通过对参数的选择进行插值曲面的局部修改,最后讨论了插值函数的一些性质.     

一类四次有理插值样条的点控制

构造了一种有理四次插值样条,其分子为四次多项式分母为二次多项式.该有理插值样条是有界的、保单调且C~2连续的,仅带有一个调节参数δ_i.研究了有理四次插值样条的性质,同时给出了相应的函数值控制、导数值控制方法,这种方法的优点在于能够根据实际设计需要简单地选取适宜的参数,达到对曲线的形状进行局部调控的目的.     

Relation between interpolating integral continued fractions and interpolating branched continued fractions

We construct and investigate an interpolating integral continued fraction, which represents a natural generalization of interpolating continued fractions. We also point out the optimal choice of the sequence of interpolating knots.     

Analysis and Construction of Multivariate Interpolating Refinable Function Vectors

In this paper, we shall introduce and study a family of multivariate interpolating refinable function vectors with some prescribed interpolation property. Such interpolating refinable function vectors are of interest in approximation theory, sampling theorems, and wavelet analysis. In this paper, we characterize a multivariate interpolating refinable function vector in terms of its mask and analyze the underlying sum rule structure of its generalized interpolatory matrix mask. We also discuss the symmetry property of multivariate interpolating refinable function vectors. Based on these results, we construct a family of univariate generalized interpolatory matrix masks with increasing orders of sum rules and with symmetry for interpolating refinable function vectors. Such a family includes several known important families of univariate refinable function vectors as special cases. Several examples of bivariate interpolating refinable function vectors with symmetry will also be presented.     

W_2~m空间中样条插值算子与最佳逼近算子的一致性

由线性微分算子确定的样条是连接多项式样条与希氏空间中抽象算子样条的重要环节,对微分算子样条的研究,既可从更高的观点揭示和概括多项式样条,又可启示我们去发现抽象算子样条的一些新的理论和应用． Ｇｒｅｅｎ函数是研究微分算子样条的重要工具 ［１］,但在微分算子插值样条的计算及将样条用于数值分析中,再生核方法起着更重要的作用．文献［２］［３］给出了与二阶线性微分算子插值样条有关的再生核解析表达式;由此得到了二阶微分算子插值样条与空间Ｗ_２～１［ａ,ｂ］中最佳插值逼近算子的一致性;而且还利用再生核给出了Ｈｉ…     

一类基于小波基函数插值的有限元方法

在分析具有大的梯度问题中,将具有紧支集的小波基函数引入到传统的有限元插值函数的构造中,对传统的插值方法进行修正。对新的插值模式进行了数值稳定性（解的唯一存在性）分析并通过分片分析讨论了解的收敛性,新的插值模式所引入的附加自由度通过静力凝聚法来消除,最后得到了基于变分原理的小波有限元列式。     

W_2~m空间中样条插值算子与最佳逼近算子的一致性

This paper discusses generalized interpolating splines which determined by n order linear differential operators, and the best operators of interpolating approximation in W_2~m spaces, The explicit constructive method for the reproducing kernel in W_2~m space is presented, and proves the uniformity of spline interpolating operators and the best operators of interpolating approximation W_2~m space by reproducing kernel. The explicit expression of approximation error on a bounded ball in W_2~m space, and error estimation of spline operator of approximation are obtained.     

连幂式插值

根据连续幂指形式的函数,提出了连续幂指形式的函数插值的概念,简称连幂式插值,用构造式方法得到了满足插值条件的连幂式插值函数。最后,通过一个算例与连分式插值函数做了对比。     

On parametrization of interpolating curves

In parametric curve interpolation there is given a sequence of data points and corresponding parameter values (nodes), and we want to find a parametric curve that passes through data points at the associated parameter values. We consider those interpolating curves that are described by the combination of control points and blending functions. We study paths of control points and points of the interpolating curve obtained by the alteration of one node. We show geometric properties of quadratic Bézier interpolating curves with uniform and centripetal parameterizations. Finally, we propose geometric methods for the interactive modification and specification of nodes for interpolating Bézier curves.