有理插值算子的连续性

1.引言 设m,n为给定的非负整数,X={z_i:z_i∈C,0≤i≤s},且z_i彼此互异。所谓有理插值问题,就是对于给定的,寻求有理函数R=P/Q∈R(m,n)(即?(P)≤m,?(Q)≤n)使得 R~(j)(z_i)=y_i~(j),j=0,1,…,k_i;i=0,1,…,s。 (1.1)而与此对应的“线性化”的问题是求P/Q∈R(m,n),使得     

一类不可微数学规划问题 Kuhn-Tucker 条件的充分性

设函数 f,g_i(i∈I={1,2,…,i})和 h_j(j∈J={1,2,…,m})均为 n 维欧氏空间 R~n 中某开集 C 上的局部 Lipschitz 函数.考虑非线性规划问题     

三次样条的Hermite-Birkboff插值

设Δ:0=x_0相似文献   

Floater和Hormann插值算子的勒贝格常数

1引言设f(x)是定义在区间[a,b]上的连续函数,插值节点x_(i)(i=0,1,…,n)满足a=x_(0)相似文献   

二元向量分叉连分式插值的矩阵算法

1 引言 设R~2中的点集Ⅱ~(n,m)由下表给出 (x_0,y_0)(x_0,y_1)…(x_0,y_m) (x_1,y_0)(x_1,y_1)…(x_1,y_m) (1.1) (x_n,y_0)(x_n,y_1)… (x_n,y_m)称Ⅱ~(n,m)为矩形网格.对Ⅱ~(n,m)中的每个点(x_i,y_i)给定d维插值向量v_(ij)并将其按上述方式排成向量网格且用中V~(n,m)记之. d维复向量V的Samelson逆定义为     

求解多项式特征值问题的部分正交投影方法及其变形

正1引言为表述方便,用C~(m×n)表示m×n复矩阵的全体,C~m=C~(m×1).‖·‖表示向量或矩阵的2-范数.对A∈C~(m×n),v∈C~m及正整数m,K[A,v,m]=[v,Av,A~2v,...,A~(m-1)v]称为Krylov矩阵,span(K[A,v,m])就是由A和v生成的Krylov子空间.e_j是适当阶单位矩阵的第j列.设A_i∈C~(m×n)(i=0,1,…,d)是给定的矩阵,记     

Hermite样条对可微函数类逼近的准确估值

1.设m为任意非负整数.以C~m表示[0,1]上具有m次连续导数的全体函数组成的集(C~0=C). 设n为正整数.以Δn表示区间[0,1]的n节分割 0=x_(0,n)相似文献   

用Walsh-Fourier级数的(c,β)平均来逼近函数的问题

设p≥2是固定的整数.x∈[0,1]的p进表示是x=(0.x_1x_2…x_n…),其中x_k∈{0,1,…,p-1},k∈N={1,2,…}。並且约定对p进有理点取有限表示。对任意非负整数k≥0,写k=sum from j=0 to n (k_jp~j),k_j∈{0,1,…,p-1}。设,则p进的Walsh函数定义为。     

关于Varma的缺插值样条函数

最近,A.K.Varma在中讨论了五次、六次缺插值样条函数。 设n=2m 1,x_i=i/2m,i=0,2,…,2m.用S_(nō)~(2)(x)表示在[0,1]上满足下列条件的五次样条函数     

Q型插值样条逼近度的精确估计

设 △={0=x_0相似文献   

INTERPOLATION BY LOOP''''S SUBDIVISION FUNCTIONS

For the problem of constructing smooth functions over arbitrary surfaces from discretedata,we propose to use Loop's subdivision functions as the interpolants.Results on theexistence,uniqueness and error bound of the interpolants are established.An efficientprogressive computation algorithm for the interpolants is also presented.     

Statistical application of barycentric rational interpolants: an alternative to splines

Spline curves, originally developed by numerical analysts for interpolation, are widely used in statistical work, mainly as regression splines and smoothing splines. Barycentric rational interpolants have recently been developed by numerical analysts, but have yet seen very few statistical applications. We give the necesssary information to enable the reader to use barycentric rational interpolants, including a suggestion for a Bayesian prior distribution, and explore the possible statistical use of barycentric interpolants as an alternative to splines. We give the all the necessary formulae, compare the numerical accuracy to splines for some Monte-Carlo datasets, and apply both regression splines and barycentric interpolants to two real datasets. We also discuss the application of these interpolants to data smoothing, where smoothing splines would normally be used, and exemplify the use of smoothing interpolants with another real dataset. Our conclusion is that barycentric interpolants are as accurate as splines, and no more difficult to understand and program. They offer a viable alternative methodology.     

INTERPOLATION BY LOOP＇S SUBDIVISION FUNCTIONS

For the problem of constructing smooth functions over arbitrary surfaces from discretedata, we propose to use Loop‘s subdivision functions as the interpolants. Results on theexistence, uniqueness and error bound of the interpolants are established. An efficientprogressive computation algorithm for the interpolants is also presented.     

On Rational Interpolation to Meromorphic Functions in Several Variables

In this paper, a new approach to construct rational interpolants to functions of several variables is considered. These new families of interpolants, which in fact are particular cases of the so-called Padé-type approximants (that is, rational interpolants with prescribed denominators), extend the classical Padé approximants (for the univariate case) and provide rather general extensions of the well-known Montessus de Ballore theorem for several variables. The accuracy of these approximants and the sharpness of our convergence results are analyzed by means of several examples.     

A Class of $C^1$ Discrete Interpolants over Tetrahedra

1.IntroductionThepurposeofthepaperistocharacterizeacertainclassofCIdiscreteinterpolalltsdefinedovertetrahedrawithonlyCIdatarequired.Weassumethatapolyhedraldomaininthree-spaceorasetof3Dscattereddatahavebeentessellatedintotetrahedrawithanytwoofwhichshareonlyoneface.Asforthispreprocessingstage,onemadreferto[2]and[3]andhereweomitit.Inthepaperlweonlydescribethecharacterizationofaninterpolantoverasingletetrahedronfortheinterpolantshavethesameform.Nowwebeginourpaperwithsomeconceptionsandnotations.…     

Multivariate refinable Hermite interpolant

We introduce a general definition of refinable Hermite interpolants and investigate their general properties. We also study a notion of symmetry of these refinable interpolants. Results and ideas from the extensive theory of general refinement equations are applied to obtain results on refinable Hermite interpolants. The theory developed here is constructive and yields an easy-to-use construction method for multivariate refinable Hermite interpolants. Using this method, several new refinable Hermite interpolants with respect to different dilation matrices and symmetry groups are constructed and analyzed.

Some of the Hermite interpolants constructed here are related to well-known spline interpolation schemes developed in the computer-aided geometric design community (e.g., the Powell-Sabin scheme). We make some of these connections precise. A spline connection allows us to determine critical Hölder regularity in a trivial way (as opposed to the case of general refinable functions, whose critical Hölder regularity exponents are often difficult to compute).

While it is often mentioned in published articles that ``refinable functions are important for subdivision surfaces in CAGD applications", it is rather unclear whether an arbitrary refinable function vector can be meaningfully applied to build free-form subdivision surfaces. The bivariate symmetric refinable Hermite interpolants constructed in this article, along with algorithmic developments elsewhere, give an application of vector refinability to subdivision surfaces. We briefly discuss several potential advantages offered by such Hermite subdivision surfaces.

     

Unified Representations of Nonlinear Splines

Golomb and Jerome's framework is modified and extended. The new framework is more general since it also handles interpolants which are not allowed to “slide” at the nodes. The space of interpolants of variable length is shown to be a smooth manifold. If the length is fixed, and there are no nodes, then the space of interpolants is a manifold. When there is at least one node, and at least one node is not on the line segment between the endpoints, then the space of interpolants of fixed length is a smooth manifold. Sufficient conditions are given which ensure the space of interpolants continues to be a smooth manifold in the presence of additional constraints such as clamping and pinning. A new fundamental finite-dimensional equation is derived. When it is solved it yields all nonlinear splines, and every nonlinear spline appears in this way. An important feature is that the same symbolic equation is used for all possible combinations of the constraints considered. It is shown how to take the solutions of the fundamental equation and use them to express the corresponding nonlinear splines in terms of a pair of elliptic functions. An inequality is derived that specifies which elliptic function appears along each section of the spline. The nonlinear splines are in a unified way shown to beC²for all possible combinations of the constraints considered.     

矩形网格上二元向量有理插值的对偶性

矩形网格上二元向量有理插值的对偶性朱功勤，檀结庆（合肥工业大学）ＴＨＥＤＵＡＬＩＴＹＯＦＢＩＶＳＲＩＡＴＥＶＥＣＴＯＲＶＡＬＵＥＤＲＡＴＩＯＮＡＬＩＮＴＥＲＰＯＬＡＮＴＳＯＶＥＲＲＥＣＴＡＮＧＵＬＡＲＧＲＩＤＳ￥ＺｈｕＧｏｎｇ－ｑｉｎ；ＴａｎＪｉｅ－...     

Positivity of cubic polynomials on intervals and positive spline interpolation

A criterion for the positivity of a cubic polynomial on a given interval is derived. By means of this result a necessary and sufficient condition is given under which cubicC ¹-spline interpolants are nonnegative. Further, since such interpolants are not uniquely determined, for selecting one of them the geometric curvature is minimized. The arising optimization problem is solved numerically via dualization.     

向量值有理插值存在性的一种判别方法

对于向量值有理插值的计算,目前已经有多种求解算法.但其存在性的判别方法及其证明在现有的文献中还没有见到.这里利用标量有理插值函数插值存在性的思想,引入Newton基函数,给出并证明了向量值有理插值存在性的一种判别方法.同时给出有理插值函数的分子和分母的显式表达式,最后的实例说明了它的有效性.