样条空间S^13（Δ^（1）mn）上的插值与逼近

本文讨论样条空间Ｓ＾１３上的插值问题，导出了一类插值条件下样条插值的存在性与唯一性结论以及计算插值样条的递推格式，其主要结论是对四阶光滑的函数，插值样条可达２阶逼近度。     

关于一类S~（1，1）_3（△~（2）_（mn））插值与逼近

设△￣（２）＿（ｍｘ）是矩形域Ｄ＝［ａ，ｂ］［ｃ，ｄ］的Ⅱ－型三角剖分．Ｓ￣（１，１）＿３（△￣（２）＿（ｍｘ）是带边界条件的三元三次样条空间：本文我们将讨论一类Ｓ￣（１，１）＿３（△￣（２）＿（ｍｘ））的插值问题，证明了它的存在性，唯一性及逼近阶：如果ｆ∈Ｃ￣４（Ｄ），则有｜ｆ－ｓ｜≤ｋ（ｌ）．ｍａｘ（ρ△，ρ￣（－１）＿△）．‖ｆ‖．．ｈ￣２．     

样条空间S~μ_k（△）的维数

本文考虑了欧式空间Ｒ￣ｎ中任意单纯形剖分上的样条函数空间．证明了当ｋ≥（３μ＋１）２￣（ｎ－２）＋１时，计算任意单纯形剖分Δ上的ｋ次μ阶光滑样条空间的维数，可归结为计算每个σ－关联域（ｉ－单纯形σ∈Δ）Ｒ（σ）上的２￣（ｎ－ｉ－１）μ次μ阶光滑（ｉ≤ｎ－１）样条空间的维数。这里σ－关联域Ｒ（σ）是指Δ中所有包含σ的单纯形所成的单纯形剖分．     

关于阶为2~（α1）3~（α2）5~（α3）7~（α4）p~（α5）的单群

本文证明了以下结果：设α１，α２，α３，α４，α５均为正整数，ｐ为素数且．如果Ｇ是阶为的单群，则Ｇ同构于下列单群之一：Ａ１１，Ａ１２；Ｍ２２，Ｈｉ－Ｓ２ＭｃＬ，Ｈｅ；Ａ１（ｑ）（ｑ＝２６，５３，７４，２９，４１，７１，２５１，４４９，４８０１），Ａ２（３２），Ａ３（２２），Ａ３（７），Ａ４（２），Ａ５（２），Ｂ２（２３），Ｂ２（７２），Ｂ３（３），Ｂ４（２），Ｃ３（３），Ｄ４（３），Ｇ２（２），Ｇ２（５），２Ａ２（１９），２Ａ３（５），２Ａ３（７），２Ａ４（３），２Ａ５（２），２Ｄ４（２）．     

关于一类S1，13（△（2）mn）插值与逼近

设△^（2）_mn是矩形域Ｄ＝［ａ，ｂ］(?)［ｃ，ｄ］的Ⅱ－型三角剖分．S^1，1₃（△^（2）_mn）是带边界条件的二元三次样条空间：本文我们将讨论一类S^1，1₃（△^（2）_mn）的插值问题，证明了它的存在性，唯一性及逼近阶：如果ｆ∈Ｃ^４（Ｄ），则有｜ｆ－ｓ｜≤ｋ（ｌ）·ｍａ     

GM（1，1）模型的样条插值函数的残差拟合

本文用样条函数对GM(1，1)模型的残差序列进行插值拟合，然后作用于二阶线性微分方程，并以此修正原模型，得到一种新的预测模型的数值解．     

关于非均匀剖分下多元样条空间S_2~1(△_(mn)~(2))的拟插值算子

１引言随着计算机科学技术的发展,多元样条在力学和计算机辅助几何设计（ＣＡＧＤ）中的应用越来越引起人们极大兴趣．然而,由于一般剖分下样条空间的研究有相当的难度,迄今为止只对于一些特殊剖分的样条空间取得了一定的进展,如：矩形剖分,均匀的１－型,２－型三角剖分等．王仁宏和崔锦泰讨论了均匀２－型三角剖下的拟插值算子以及其逼近性质,鉴于在工程和实际应用中均匀剖分具有一定局限性,作者在文献（［１］,［３］）的基础上,对于非均匀２－型三角剖分,给出了一类拟插值算子,并研究了它的逼近性质．同时,利用其构造了一类…     

渐近Fejer点上（0，1，…，q）Hermite—Fejer插值多项式的逼近阶

本文得到了渐近Fejer点上的（0,1,…,q)Hermite-Fejer插值多项式在边界有二阶连续导数的区域D上平均逼近函数类A（－↑D）中被插值函数的逼近阶，同时还得到了在D上的一致逼近的逼近阶，并指出逼近阶是精确的。     

单位根上重Hermite插值多项式的逼近阶

设ｆ（ｚ）在｜ｚ｜≤１解析，在｜ｚ｜≤１连续．本文得到了基于单位根的扩充Ｈｅｒｍｉｔｅ插值多项式在｜ｚ｜≤１上一致收敛于ｆ（ｚ）的逼近阶和在｜ｚ｜＝１上平均收敛于ｆ（ｚ）的逼近阶，且一般说来，阶还是精确的．进而说明，重数不同的插值多项式的逼近阶不同于重数相同的插值多项式的逼近阶．     

使用基样条构造多分辨率逼近及小波

我们首先介绍了Ｂ－样条及基样条，然后用ｍ阶的Ｂ－样条Ｎｍ（ｘ）生成一个Ｌ＾２（Ｒ）中一个比例为ｒ的多分辨逼近，而且用（ψｔ（ｘ）＝Ｌ＾（ｍ）２ｍ（ｒｘ－ｔ），ｔ＝１，２，．．．ｘ－１）构造了相应的小波空间，这里Ｌ２ｍ为２ｍ阶的基样条，最后，我们给出了小波的分解与合成算法。     

凸四边形上一类具有C~2-连续的插值问题的样条解及其应用

本文讨论了一类凸四边形上的插值问题.指出这类插值问题是可解的,其解是分片二元三次多项式,且在凸四边形上是C~2-连续的.我们证明了这类插值问题的解的存在性和唯一性,给出了解样条的分片表达式及其逼近度的估计.最后还给出了一个应用实例和图形显示来说明本方法是可行的.     

Interpolation and Approximation by Splines on [0, ∞ )

We investigate interpolation and approximation problems by splines, which possess a countable set of knots on the positive axis. In particular, we characterize those sets of points, which admit unique Lagrange interpolation and give some sufficient and some necessary conditions for best approximations. Moreover, we show that the classical results of spline-approximation theory are not available for splines with a countable set of knots.     

On the cardinal spline interpolation corresponding to infinite order differential operators

This paper discusses some problems on the cardinal spline interpolation corresponding to infinite order differential operators. The remainder formulas and a dual theorem are established for some convolution classes, where the kernels arePF densities. Moreover, the exact error of approximation of a convolution class with interpolation cardinal splines is determined. The exact values of averagen-Kolmogorov widths are obtained for the convolution class. Supported in part by NSFC.     

Some extremal properties of multivariate polynomial splines in the metric Lp (Rd )

We constructed a kind of continuous multivariate spline operators as the approximation tools of the multivariate functions on the Bd instead of the usual multivariate cardinal interpolation oper-ators of splines, and obtained the approximation error by this kind of spline operators. Meantime, by the results, we also obtained that the spaces of multivariate polynomial splines are weakly asymptoti-cally optimal for the Kolmogorov widths and the linear widths of some anisotropic Sobolev classes of smooth functions on Bd in the metric Lp(Bd).     

Lagrange Interpolation by Bivariate C 1-Splines with Optimal Approximation Order

We develop the first local Lagrange interpolation scheme for C ¹-splines of degree q≥3 on arbitrary triangulations. For doing this, we use a fast coloring algorithm to subdivide about half of the triangles by a Clough–Tocher split in an appropriate way. Based on this coloring, we choose interpolation points such that the corresponding fundamental splines have local support. The interpolating splines yield optimal approximation order and can be computed with linear complexity. Numerical examples with a large number of interpolation points show that our method works efficiently.     

Local Lagrange Interpolation with Bivariate Splines of Arbitrary Smoothness

We describe a method which can be used to interpolate function values at a set of scattered points in a planar domain using bivariate polynomial splines of any prescribed smoothness. The method starts with an arbitrary given triangulation of the data points, and involves refining some of the triangles with Clough-Tocher splits. The construction of the interpolating splines requires some additional function values at selected points in the domain, but no derivatives are needed at any point. Given n data points and a corresponding initial triangulation, the interpolating spline can be computed in just O(n) operations. The interpolation method is local and stable, and provides optimal order approximation of smooth functions.     

A local Lagrange interpolation method based on cubic splines on Freudenthal partitions

A trivariate Lagrange interpolation method based on cubic splines is described. The splines are defined over a special refinement of the Freudenthal partition of a cube partition. The interpolating splines are uniquely determined by data values, but no derivatives are needed. The interpolation method is local and stable, provides optimal order approximation, and has linear complexity.

     

Approximation order of bivariate spline interpolation for arbitrary smoothness

By using the algorithm of Nürnberger and Riessinger (1995), we construct Hermite interpolation sets for spaces of bivariate splines S_q^r(Δ¹) of arbitrary smoothness defined on the uniform type triangulations. It is shown that our Hermite interpolation method yields optimal approximation order for q 3.5r + 1. In order to prove this, we use the concept of weak interpolation and arguments of Birkhoff interpolation.     

Semi-cardinal interpolation and difference equations: From cubic B-splines to a three-direction box-spline construction

This paper considers the problem of interpolation on a semi-plane grid from a space of box-splines on the three-direction mesh. Building on a new treatment of univariate semi-cardinal interpolation for natural cubic splines, the solution is obtained as a Lagrange series with suitable localization and polynomial reproduction properties. It is proved that the extension of the natural boundary conditions to box-spline semi-cardinal interpolation attains half of the approximation order of the cardinal case.     

Bivariate spline interpolation at grid points

Summary. We describe algorithms for constructing point sets at which interpolation by spaces of bivariate splines of arbitrary degree and smoothness is possible. The splines are defined on rectangular partitions adding one or two diagonals to each rectangle. The interpolation sets are selected in such a way that the grid points of the partition are contained in these sets, and no large linear systems have to be solved. Our method is to generate a net of line segments and to choose point sets in these segments which satisfy the Schoenberg-Whitney condition for certain univariate spline spaces such that a principle of degree reduction can be applied. In order to include the grid points in the interpolation sets, we give a sufficient Schoenberg-Whitney type condition for interpolation by bivariate splines supported in certain cones. This approach is completely different from the known interpolation methods for bivariate splines of degree at most three. Our method is illustrated by some numerical examples. Received October 5, 1992 / Revised version received May 13, 1994