Nonconforming rotated Q 1 element on non-tensor product anisotropic meshes

In this paper, we consider the nonconforming rotated Q ₁ element for the second order elliptic problem on the non-tensor product anisotropic meshes, i.e. the anisotropic affine quadrilateral meshes. Though the interpolation error is divergent on the anisotropic meshes, we overcome this difficulty by constructing another proper operator. Then we give the optimal approximation error and the consistency error estimates under the anisotropic affine quadrilateral meshes. The results of this paper provide some hints to derive the anisotropic error of some finite elements whose interpolations do not satisfy the anisotropic interpolation properties. Lastly, a numerical test is carried out, which coincides with our theoretical analysis.     

Weighted approximation of functions on the real line by Bernstein polynomials

The authors give error estimates, a Voronovskaya-type relation, strong converse results and saturation for the weighted approximation of functions on the real line with Freud weights by Bernstein-type operators.     

Multivariate approximation in the worst case setting over reproducing kernel Hilbert spaces

We study the worst case setting for approximation of d variate functions from a general reproducing kernel Hilbert space with the error measured in the L_∞ norm. We mainly consider algorithms that use n arbitrary continuous linear functionals. We look for algorithms with the minimal worst case errors and for their rates of convergence as n goes to infinity. Algorithms using n function values will be analyzed in a forthcoming paper.We show that the L_∞ approximation problem in the worst case setting is related to the weighted L₂ approximation problem in the average case setting with respect to a zero-mean Gaussian stochastic process whose covariance function is the same as the reproducing kernel of the Hilbert space. This relation enables us to find optimal algorithms and their rates of convergence for the weighted Korobov space with an arbitrary smoothness parameter α>1, and for the weighted Sobolev space whose reproducing kernel corresponds to the Wiener sheet measure. The optimal convergence rates are n^-(α-1)/2 and n^-1/2, respectively.We also study tractability of L_∞ approximation for the absolute and normalized error criteria, i.e., how the minimal worst case errors depend on the number of variables, d, especially when d is arbitrarily large. We provide necessary and sufficient conditions on tractability of L_∞ approximation in terms of tractability conditions of the weighted L₂ approximation in the average case setting. In particular, tractability holds in weighted Korobov and Sobolev spaces only for weights tending sufficiently fast to zero and does not hold for the classical unweighted spaces.     

Pointwise Simultaneous Approximation by Combinations of Baskakov Operators

In this paper, we investigate the relation between the rate of convergence for the derivatives of the combinations of Baskakov operators and the smoothness for the derivatives of the functions approximated. We give some direct and inverse results on pointwise simultaneous approximation by the combinations of Baskakov operators. We also give a new equivalent result on pointwise approximation by these operators.     

一组三个函数乘积的高阶导数公式及其行列式表示

研究了三个函数乘积的高阶导数,得到了一组相应的导数公式.利用这些公式求该类函数的高阶导数以及进行近似计算,或做函数的近似表示,将起到较好的简化作用.     

Neumann—Bessel级数的核函数的渐近表示及其Fejer和的类逼近

先给出Neumann-Bessel级数的核函数的精确的渐近表示,然后讨论该级数的部分和的收敛速度及其Fejer和的类逼近问题,从中说明了对于非三角级数也能得到相应三角级数中类逼近的精密结果。     

On convergence rates in approximation theory for operator semigroups

We create a new, functional calculus, approach to approximation formulas for _C0$C_0$-semigroups on Banach spaces restricted to the domains of fractional powers of their generators. This approach allows us to equip the approximation formulas with rates which appear to be optimal in a natural sense. In the case of analytic semigroups, we improve our general results obtaining better convergence rates which are optimal in that case too. The setting of analytic semigroups includes also the case of convergence on the whole space. As an illustration of our approach, we deduce optimal convergence rates in classical approximation formulas for _C0$C_0$-semigroups restricted to the domains of fractional powers of their generators.     

Approximation of Vector Valued Functions

In this paper, we extend the well-known result of Lipschitz approximation of lower semi-continuous functions to a class of lattice valued vector functions. We then use this approximation to get convergence results and we give two applications to the law of large numbers and to an ergodic theorem.     

On the Best Choice for the Generating Polynomial of a Padé-Type Approximation

The first aim of this paper is the study and determination of the best choices (pointwise, L ² and uniform) for the generating polynomial of a Padé-type approximation to the Taylor series of a function analytic in an open planar disk. The second aim is to characterize the corresponding Hermite polynomial and to give some estimates for the uniform norm of a Padé-type approximation error.     

INTERPOLATION INEQUALITIES AND DECAY ESTIMATES FOR DERIVATIVES

Under the only assumption of the cone property for a given domain Rn, it is proved that inter polation inequalities for intermediate derivatives of functions in the Sobolev spaces Wm,p or even in some weighted Sobolve spaces still hold. That is, the usual additional restrictions that is bounded or has the uniform cone property are both removed. The main tools used are polynomial inequalities, by which it is also ob- tained pointwise version inter polation inequalities for smooth and analytic functions. Such pointwise version in - equalities give explicit decay estimates for derivatives at infinity in unbounded domains which have the cone property. As an application of the decay ertimates, a previous result on radial basis function approximation of smooth functions is extended to the derivative-simultaneous approximation.     

The complexity of anl p method for discrete Tchebycheff approximation in exact arithmetic

A discrete Tchebycheff problem is approximated by a sequence ofl _p norm problems. This is an algorithm to be used if a large number of variables is involved as is the case in the approximation of a function of several variables by a polynomial. The complexity of this procedure is investigated and a lower bound for the number of steps to reach ε-optimality is established. Supported by NIH Grant RR01243 at the University of Washington     

一种新的快速逼近真值的程序算法──分布误差法

本文介绍一种新的逼近真值的快速程序算法及其应用。     

On a constructive approximation of the efficient outcomes in bicriterion vector optimization

For bicriterion quasiconvex optimization problems, we present a constructive procedure for an approximation of the efficient outcomes. Performing this procedure we can estimate the accuracy of the approximation. Conversely, if we prescribe an accuracy for the approximation, we can calculate the number of points which have to be computed by a certain scalarization method to remain under the given accuracy. Finally, we give a numerical example.     

Localization of Frames

Motivated by Balan, Casazza, Heil, and Landau's results on localization of frames in a separable Hilbert space ?, we investigate the transitivity of localization relation of the frame ? = {f _i } _i∈I and sequence ? = {e _j } _j∈G in ? with respect to the associated map a: I → G, where G is a discrete abelian group and I is a index set. We also study some properties of ? _p -column decay, ? _p -row decay, weak homogeneous approximation property, and strong homogeneous approximation property of frame ? = {f _i } _i∈I , and sequence ? = {e _j } _j∈G , with respect to the associated map a.     

二元Lagrange三角插值多项式在Orlicz空间内的逼近

研究了二元函数用一种组合型的三角插值多项式算子逼近的问题.借助连续模这一工具,给出了这类三角插值多项式在Orlicz空间内的逼近定理.     

Enumerating Quartic Dihedral Extensions of ℚ

We give an explicit Dirichlet series for the generating function of the discriminants of quartic dihedral extensions of . From this series we deduce an asymptotic formula for the number of isomorphism classes of such quartic extensions with discriminant up to a given bound. On the other hand, by using essentially classical results of genus theory combined with elementary analytical methods such as the method of the hyperbola, we show how to compute exactly this number up to quite large bounds, and we give a table of selected values.     

Banach空间中线性算子的Drazin广义逆的表示

1981年,1985年,2000年,乔三正、蔡东汉、魏益民分别给出Drazin广义逆不同形式的表示．本文将对上述结果进行推广,给出Drazin广义逆的统一表示,使上述三个结果均成为本文主要结果的特例．在本文主要结果的基础上,利用算子谱理论,给出Drazin广义逆的一种逼近形式的表示,同时给出逼近解的估计．此结果推广了蔡东汉、魏益民的相应结果．     

An extension based on qR-integral for a sequence of operators

The paper deals with a sequence of linear positive operators introduced via q-Calculus. We give a generalization in Kantorovich sense of its involving qR-integrals. Both for discrete operators and for integral operators we study the error of approximation for bounded functions and for functions having a polynomial growth. The main tools consist of the K-functional in Peetre sense and different moduli of smoothness.     

The generalization of Meyer-König and Zeller operators by generating functions

In this paper, theorems are proved concerned with some approximation properties of generating functions type Meyer-König and Zeller operators with the help of a Korovkin type theorem. Secondly, we compute the rates of convergence of these operators by means of the modulus of continuity, Peetre's K-functional and the elements of the modified Lipschitz class. Also we introduce the rth order generalization of these operators and we obtain approximation properties of them. In the last part, we give some applications to the differential equations.