二元三次样条空间S31,2(Δmn(2)) 的样条拟插值

本文首先利用由两组具有局部最小支集的样条所组成的基函数,构造非均匀2 型三角剖分上二元三次样条空间S₃^1,2(Δ_mn⁽²⁾)的若干样条拟插值算子. 这些变差缩减算子由样条函数B_ij¹支集上5 个网格点或中心和样条函数B_ij²支集上5 个网格点处函数值定义. 这些样条拟插值算子具有较好的逼近性,甚至算子V_mn（f） 能保持近最优的三次多项式性. 然后利用连续模,分析样条拟插值算子V_mn（f）一致逼近于充分光滑的实函数. 最后推导误差估计.     

Quadratic spline quasi-interpolants on Powell-Sabin partitions

In this paper we address the problem of constructing quasi-interpolants in the space of quadratic Powell-Sabin splines on nonuniform triangulations. Quasi-interpolants of optimal approximation order are proposed and numerical tests are presented. Dedicated to Prof. Mariano Gasca on the occasion of his 60th birthday.     

A numerical method for solving KdV equation with multilevel B-spline quasi-interpolation

In this article, we use a multilevel quartic spline quasi-interpolation scheme to solve the one-dimensional nonlinear Korteweg–de Vries (KdV) equation which exhibits a large number of physical phenomena. The presented scheme is obtained by using the second-order central divided difference of the spatial derivative to approximate the third-order spatial derivative, and the forward divided difference to approximate the temporal derivative, where the spatial derivative is approximated by the proposed quasi-interpolation operator. Compared to other numerical methods, the main advantages of our scheme are the higher accuracy and lower computational complexity. Meanwhile, the algorithm is very simple and easy to implement. Numerical experiments in this article also show that our scheme is feasible and valid.     

Construction Techniques for Highly Accurate Quasi-Interpolation Operators

Under mild additional assumptions this paper constructs quasi-interpolants in the form

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with approximation order ℓ−1, where_h(x) is a linear combination of translatesψ(x−jh) of a functionψinC^ℓ( ). Thus the order of convergence of such operators can be pushed up to a limit that only depends on the smoothness of the functionψ. This approach can be generalized to the multivariate setting by using discrete convolutions with tensor products of odd-degreeB-splines.     

一种2型三角剖分上多元样条拟插值的精度提高策略

给定一个多元拟插值算子, 若其具有单位分解性质 (再生0次多项式), 我们提出一种利用其周围节点提高多项式再生性的方法. 所得算子不仅具有更高的逼近精度, 还不需要目标函数的任何导数信息. 然后利用此方法, 我们改进了2型三角剖分上的多元样条拟插值,使之具有更高的精度. 最后, 我们应用改进的拟插值算子数值求解时间发展偏微分方程. 数值实验验证了该方法的有效性.     

A Quasi-Interpolation Satisfying Quadratic Polynomial Reproduction with Radial Basis Functions

In this paper,a new quasi-interpolation with radial basis functions which satis- fies quadratic polynomial reproduction is constructed on the infinite set of equally spaced data.A new basis function is constructed by making convolution integral with a constructed spline and a given radial basis function.In particular,for twicely differ- entiable function the proposed method provides better approximation and also takes care of derivatives approximation.     

Approximate approximations from scattered data

The aim of this paper is to extend the approximate quasi-interpolation on a uniform grid by dilated shifts of a smooth and rapidly decaying function to scattered data quasi-interpolation. It is shown that high order approximation of smooth functions up to some prescribed accuracy is possible, if the basis functions, which are centered at the scattered nodes, are multiplied by suitable polynomials such that their sum is an approximate partition of unity. For Gaussian functions we propose a method to construct the approximate partition of unity and describe an application of the new quasi-interpolation approach to the cubature of multi-dimensional integral operators.     

Generator,multiquadric generator,quasi-interpolation and multiquadric quasi-interpolation

The aim of this survey paper is to propose a new concept “generator”. In fact, generator is a single function that can generate the basis as well as the whole function space. It is a more fundamental concept than basis. Various properties of generator are also discussed. Moreover, a special generator named multiquadric function is introduced. Based on the multiquadric generator, the multiquadric quasi-interpolation scheme is constructed, and furthermore, the properties of this kind of quasi-interpolation are discussed to show its better capacity and stability in approximating the high order derivatives.     

Affine Synthesis onto L p when 0<p≤1

The affine synthesis operator is shown to map the coefficient space ℓ ^p (ℤ₊×ℤ ^d ) surjectively onto L ^p (ℝ ^d ), for p∈(0,1]. Here ψ _j,k (x)=|det a _j |^1/p ψ(a _j x−k) for dilation matrices a _j that expand, and the synthesizer ψ∈L ^p (ℝ ^d ) need satisfy only mild restrictions, for example, ψ∈L ¹(ℝ ^d ) with nonzero integral or else with periodization that is real-valued, nontrivial and bounded below. An affine atomic decomposition of L ^p follows immediately:

基于双二次样条函数的一类保形拟插值

给定一组数据点{(xi,yj,f(xi,yj)}(i=1,2,…,n;j=1,2,…,m)构造一类由双二次样条函数生成的保形拟插值σ(x,y)=(n+k∑i=1-k)(m+l∑j=1-lf)(xi,yj)Bkl ij(x,y),1≤k≤n,1≤l≤m,证明了σ(x,y)具有线性再生性,并且保持原有数据点的单调性和凸性等一系列保形性质.在计算机辅助几何设计和曲线曲面造型技术中利用这一性质设计和构造曲线或者曲面是相当便利的.