T样条函数空间的局部支集基

Schumaker，L.L.在其名著《Spline Function；Basic Theory》中第九章给出了Tchebysheff样条函数空间的局部支集基定理，可惜其证明却是错的，本文给出了上述定理的正确证明。     

对称性与局部支集样条函数构造

1引言令D为二维串连通区域,用不可约代数曲线对D进行剖分西,得胞腔。,i—l,…,N．D上n次广阶光滑的样条函数空间定义为S：（D,凸）一｛f6or（D）,f。一片;6尺｝在两相邻胞腔。和个上,SESZ（D,凸）满足其中l。为人与4的公共内网线,q。称为S在l。上的光滑余因子．进而分片多项式S6s：（,凸）,当且仅当在任一内网线上存在光滑余因子,且在任一内网点A处满足协调条件Zc／xx。一。其中求和对所有以A为一端点的内网线进行［”‘j．以上定义和基本结果是文1中给出的．这种方法称为光滑余因子协条法．此外在多元样条函数的研究…     

构造非均匀剖分上局部支集样条函数的积分方法

１　引　　言 给定递增数列｛ｘｉ｝,｛ｙｉ｝,｛ｚｋ｝．对于Ｒ２,由直线｛ｘ＝ｘｉ）和｛ｙ＝ｙｊ｝形成的剖分称为非均匀矩形剖分;再连接每个矩形［ｘｉ,ｘｉ＋１］×［ｙｊ,ｙｊ＋１］的正斜率对角线形成的剖分称为非均匀Ｉ－型三角剖分,记为　１;若连接每个矩形的两条对角线,则所得剖分为非均匀Ⅱ－型三角剖分,记为　２．对于Ｒ３,由平面｛ｘ＝ｘｉ｝,｛ｙ＝ｙｊ｝,｛ｚ＝ｚｋ｝形成的剖分称为非均匀长方体剖分;若再连接每个小长方体［ｘｉ,ｘｉ＋１］×［ｙｊ,ｙｊ＋１］×［ｚｋ,ｚｋ＋１］的过点（ｘｉ,ｙｊ,ｚ…     

最小支集样条小波有限元

本文认真分析研究了最小支集样条小波及其有关性质,用以张量积形式构造的二维小波建立了最小支集样条小波插值函数,讨论了其相关的性质,随后用最小支集样条小波有限元法去解弹性薄板小挠度问题,给出了数值解的误差阶,最后列举了一个数值例子．     

带边界条件的二元样条函数空间

§1.引言 本文主要讨论某一类带边界条件的二元样条函数空间的维数及其局部基函数.这方面的工作见[1—3]. 设Ω=[0,k+1]×[0,?+1].记△_(k?)~1是Ω上的三方向分划(见图1),△_(k?)~2是Ω上的四方向分划(见图2).     

紧支集分布和可加细向量的整平移的线性相关性

In this paper, the global and local linear independence of any compactly supported distributions by using time domain spaces ,and of refinable vectors by invariant linear spaces are investigated.     

加细剖分样条函数空间的维数级数和基函数

在文[1]中,我们讨论了利用协调矩阵计算维数级数和基函数的Grobner基方法,本文考虑几种加细剖分样条函数空间的维数级数和基函数,给出了它们的表达式。 1 任意三角剖分的连续样条函数 任意三角剖分上连续样条函数空间的维数早已被确定。本节我们用Grobner基方法来计算其维数级数的发生函数和基函数。 设Δ是单连通区域D上的三角剖分,f_0~0(Δ)是Δ内点的个数,f_1~0(Δ)是内网线的个数,f_2~0(Δ)是三角形的个数,我们有     

加密网格点二元局部基插值样条函数

1.简介 由于在理论以及应用两方面的重要性,多元样条引起了许多人的注意([6],[7]),紧支撑光滑分片多项式函数对于曲面的逼近是一个十分有效的工具。由于它们的局部支撑性,它们很容易求值;由于它们的光滑性,它们能被应用到要满足一定光滑条件的情况下;由于它们是紧支撑的,它们的线性包有很大的逼近灵活性,而且用它们构造逼近方法来解决的系统是     

四个方向网上二元样条的最小支集性

本文利用截断函数讨论了四个方向网上二元样条的最小支集性。我们确定了样条空间中的最小支集元,并证明了它们构成紧支集样条空间的一组基。     

Bivariate Polynomial Natural Spline Interpolation Algorithms with Local Basis for Scattered Data

Because of its importance in both theory and applications, multivariate splines have attracted special attention in many fields. Based on the theory of spline functions in Hilbert spaces, bivariate polynomial natural splines for interpolating, smoothing or generalized interpolating of scattered data over an arbitrary domain are constructed with one-sided functions. However, this method is not well suited for large scale numerical applications. In this paper, a new locally supported basis for the bivariate polynomial natural spline space is constructed. Some properties of this basis are also discussed. Methods to order scattered data are shown and algorithms for bivariate polynomial natural spline interpolating are constructed. The interpolating coefficient matrix is sparse, and thus, the algorithms can be easily implemented in a computer.     

Local Accuracy for Radial Basis Function Interpolation on Finite Uniform Grids

We consider interpolation on a finite uniform grid by means of one of the radial basis functions (RBF) φ(r)=r^γ for γ>0, γ2 or φ(r)=r^γ ln r for γ2 ₊. For each positive integer N, let h=N⁻¹ and let {x_i: i =1, 2, …, (N+1)^d} be the set of vertices of the uniform grid of mesh-size h on the unit d-dimensional cube [0, 1]^d. Given f: [0, 1]^d→ , let s_h be its unique RBF interpolant at the grid vertices: s_h(x_i)=f(x_i), i=1, 2, …, (N+1)^d. For h→0, we show that the uniform norm of the error f−s_h on a compact subset K of the interior of [0, 1]^d enjoys the same rate of convergence to zero as the error of RBF interpolation on the infinite uniform grid h ^d, provided that f is a data function whose partial derivatives in the interior of [0, 1]^d up to a certain order can be extended to Lipschitz functions on [0, 1]^d.     

On the Hilbert Transform in Bergman Space

In this paper we describe the image of the Hilbert transform operator for Bergman space.     

Construction of Continuous Functions with Prescribed Local Regularity

In this paper we investigate from both a theoretical and a practical point of view the following problem: Let s be a function from [0;1] to [0;1] . Under which conditions does there exist a continuous function f from [0;1] to R such that the regularity of f at x , measured in terms of H?lder exponent, is exactly s(x) , for all x ∈ [0;1] ? We obtain a necessary and sufficient condition on s and give three constructions of the associated function f . We also examine some extensions regarding, for instance, the box or Tricot dimension or the multifractal spectrum. Finally, we present a result on the ``size' of the set of functions with prescribed local regularity. November 30, 1995. Date revised: September 30, 1996.     

球面上径向基函数最小范数插值逼近的误差估计

研究了球面径向基插值对球面函数的逼近问题,给出了一致逼近的上界估计式．文中结果说明,球面径向基插值的逼近阶会随函数光滑性的提高而增加．     

欧氏空间中等角基的性质

等角基是正交基的推广,等角基具有和正交基相似的性质,因此研究等角基的性质能够为研究欧氏空间提供一种工具,加深对欧氏空间的了解.本文主要把n维欧氏空间中正交基的一些性质推广到等角基上,得到了五个关于等角基性质的定理.     

Factorization of Operator Functions in a Hilbert Space

Let be an integer. A Fermat-Euler dynamical system acts on the set of mod- residues coprime to by multiplication by a constant (which is also coprime to ). We study the dependence of the period and the number of orbits of this dynamical system on . Theorems generalizing Fermat's little theorem, as well as empirical conjectures, are given.     

欧氏空间中等角基的性质

等角基是正交基的推广,等角基具有和正交基相似的性质,因此研究等角基的性质能够为研究欧氏空间提供一种工具,加深对欧氏空间的了解．本文主要把n维欧氏空间中正交基的一些性质推广到等角基上,得到了五个关于等角基性质的定理．     

On the Support of Distributions of Integer-Valued Additive Functions

We obtain a characterization of a set of additive functions with asymptotically finite support. Additive functions taking integer values on prime powers p ^r are considered.     

Sobolev-Type Classes of Functions with Values in a Metric Space. II

We prove equivalence of the definitions by the author and by Korevaar and Schoen of the Sobolev classes of mappings of a domain of an arithmetic n-dimensional space to a metric space.