Local Approximation from Spline Spaces on Box Meshes

Foundations of Computational Mathematics - This paper analyzes the approximation properties of spaces of piecewise tensor product polynomials over box meshes with a focus on application to...     

Explicit forms for certain Hermite approximations

The Hermite polynomial approximation in which function values and the first few derivatives of a given function are matched at two points is given in an explicit form and an error term is deduced. These results are extended to two dimensions.     

Convergent Vector and Hermite Subdivision Schemes

Hermite subdivision schemes have been studied by Merrien, Dyn, and Levin and they appear to be very different from subdivision schemes analyzed before since the rules depend on the subdivision level. As suggested by Dyn and Levin, it is possible to transform the initial scheme into a uniform stationary vector subdivision scheme which can be handled more easily.With this transformation, the study of convergence of Hermite subdivision schemes is reduced to that of vector stationary subdivision schemes. We propose a first criterion for C⁰-convergence for a large class of vector subdivision schemes. This gives a criterion for C¹-convergence of Hermite subdivision schemes. It can be noticed that these schemes do not have to be interpolatory. We conclude by investigating spectral properties of Hermite schemes and other necessary/sufficient conditions of convergence.     

SHAPE-PRESERVING APPROXIMATION BY SPLINE FUNCTIONS

The author establishes the Jackson-type estimates for monotone and convex approxi-mation by spline functions with non-equally spaced knots.These estimaes involve highorder modulus of the approximated function and its derivatives.Hence some Bernstein-type theorems can be got conveniently.     

Hermite Subdivision Schemes and Taylor Polynomials

We propose a general study of the convergence of a Hermite subdivision scheme ℋ of degree d>0 in dimension 1. This is done by linking Hermite subdivision schemes and Taylor polynomials and by associating a so-called Taylor subdivision (vector) scheme . The main point of investigation is a spectral condition. If the subdivision scheme of the finite differences of is contractive, then is C ⁰ and ℋ is C ^d . We apply this result to two families of Hermite subdivision schemes. The first one is interpolatory; the second one is a kind of corner cutting. Both of them use the Tchakalov-Obreshkov interpolation polynomial.      

Construction of Local Quartic Spline Elements for Optimal-Order Approximation

This paper is concerned with a study of approximation order and construction of locally supported elements for the space of (piecewise polynomial) functions on an arbitrary triangulation of a connected polygonal domain in . It is well known that even when is a three-directional mesh , the order of approximation of is only 4, not 5. The objective of this paper is two-fold: (i) A local Clough-Tocher refinement procedure of an arbitrary triangulation is introduced so as to yield the optimal (fifth) order of approximation, where locality means that only a few isolated triangles need refinement, and (ii) locally supported Hermite elements are constructed to achieve the optimal order of approximation.

     

Analysis and Convergence of Hermite Subdivision Schemes

Foundations of Computational Mathematics - Hermite interpolation property is desired in applied and computational mathematics. Hermite and vector subdivision schemes are of interest in CAGD for...     

On Approximation with Spline Generated Framelets

We characterize the approximation spaces associated with the best n-term approximation in L_p(R) by elements from a tight wavelet frame associated with a spline scaling function. The approximation spaces are shown to be interpolation spaces between L_p and classical Besov spaces, and the result coincides with the result for nonlinear approximation with an orthonormal wavelet with the same smoothness as the spline scaling function. We also show that, under certain conditions, the Besov smoothness can be measured in terms of the sparsity of expansions in the wavelet frame, just like the nonredundant wavelet case. However, the characterization now holds even for wavelet frame systems that do not have the usually required number of vanishing moments, e.g., for systems built through the Unitary Extension Principle, which can have no more than one vanishing moment. Using these results, we describe a fast algorithm that takes as input any function and provides a near sparsest expansion of it in the framelet system as well as approximants that reach the optimal rate of nonlinear approximation. Together with the existence of a fast algorithm, the absence of the need for vanishing moments may have an important qualitative impact for applications to signal compression, as high vanishing moments usually introduce a Gibbs-type phenomenon (or ringing artifacts)in the approximants.     

On Implementation of Non-Polynomial Spline Approximation

Computational Mathematics and Mathematical Physics - In this paper, different variants of processing of number flows using Lagrange and Hermite non-polynomial splines are studied. The splines are...     

渐近于Hermite多项式的双正交系统

本文利用渐近于Gauss函数的函数类?,给出渐近于Hermite正交多项式的一类Appell多项式的构造方法,使得该序列与?的n阶导数之间构成了一组双正交系统.利用此结果,本文得到多种正交多项式和组合多项式的渐近性质.特别地,由N阶B样条所生成的Appell多项式序列恰为N阶Bernoulli多项式.从而,Bernoulli多项式与B样条的导函数之间构成了一组双正交系统,且标准化之后的Bernoulli多项式的渐近形式为Hermite多项式.由二项分布所生成的Appell序列为Euler多项式,从而,Euler多项式与二项分布的导函数之间构成一组双正交系统,且标准化之后的Euler多项式渐近于Hermite多项式.本文给出Appell序列的生成函数满足的尺度方程的充要条件,给出渐近于Hermite多项式的函数列的判定定理.应用该定理,验证广义Buchholz多项式、广义Laguerre多项式和广义Ultraspherical(Gegenbauer)多项式渐近于Hermite多项式的性质,从而验证超几何多项式的Askey格式的成立.     

Approximation of analytic functions by Hermite functions

We solve the inhomogeneous Hermite equation and apply this result to estimate the error bound occurring when any analytic function is approximated by an appropriate Hermite function.     

Hermite Interpolation of Data Sets by Minimal Spline Spaces

Hermite interpolation of 2n + k data by spline spaces of order k with n variable knots counting multiplicities is studied. A characterization of the minimal spline spaces which admit a solution of the interpolation problem is obtained. A sufficient condition on uniqueness of interpolating spline functions is given.     

Hermite插值算子在Orlicz空间内的逼近

插值算子逼近是逼近论中一个非常有趣的问题,尤其是以一些特殊的点为结点的插值算子的逼近问题很受人们的关注.研究了以第一类Chebyshev多项式零点为插值结点的Hermite插值算子在Orlicz范数下的逼近.     

C2 Continuous Quartic Hermite Spline Curves with Shape Parameters

In order to relieve the deficiency of the usual cubic Hermite spline curves, the quartic Hermite spline curves with shape parameters is further studied in this work. The interpolation error and estimator of the quartic Hermite spline curves are given. And the characteristics of the quartic Hermite spline curves are discussed. The quartic Hermite spline curves not only have the same interpolation and conti-nuity properties of the usual cubic Hermite spline curves, but also can achieve local or global shape adjustment and C2 continuity by the shape parameters when the interpolation conditions are fixed.     

A Class of Three-Level Explicit Difference Schemes

A class of three-level six-point explicit schemes $L_3$ with two parameters $s, p$ and accuracy $O(\tau h+h^2)$ for a dispersion equation $U_1=aU_{xxx}$ is established. The stability condition $|R|\leq 1.35756176$ $(s=3/2,p=1.184153684)$ for $L_3$ is better than $|R|$ ＜ 1.1851 in [1] and seems to be the best for schemes of the same type.     

Approximation Schemes for Functional Optimization Problems

Approximation schemes for functional optimization problems with admissible solutions dependent on a large number d of variables are investigated. Suboptimal solutions are considered, expressed as linear combinations of n-tuples from a basis set of simple computational units with adjustable parameters. Different choices of basis sets are compared, which allow one to obtain suboptimal solutions using a number n of basis functions that does not grow “fast” with the number d of variables in the admissible decision functions for a fixed desired accuracy. In these cases, one mitigates the “curse of dimensionality,” which often makes unfeasible traditional linear approximation techniques for functional optimization problems, when admissible solutions depend on a large number d of variables. Marcello Sanguineti was partially supported by a PRIN grant from the Italian Ministry for University and Research (project “Models and Algorithms for Robust Network Optimization”).     

基于系数逼近的差分格式

对于变系数微分方程,在每个离散子区间上用函数去逼近系数比用一常数去代替系数,所得到的一系列近似微分方程有更高的精度．通常的差分格式建立在解函数在子区间上的Taylor展开式的近似的基础上,这样要求函数相对于网格是缓变的．而基于系数Taylor展开的近似式和局部基的引入,使得方法能在子区间上精确表达比二次函数丰富得多的解函数．由此构造的差分格式能在子区间上反映解具有迅速变化（如边界层,高振荡）的复杂的物理现象．数值实验（边值问题、特征值问题）显示了新方法比传统方法有更满意的效果．     

Hermite插值多项式在不同基下的显式表示

本文利用对偶基的概念,导出了 Ｈｅｒｍ ｉｔｅ 插值多项式在不同基下的显式表示,这给人们对 Ｈｅｒｍ ｉｔ插值多项式在不同基下从一种表示转换到另一种表示带来极大的方便