The Chebyshev approximation of a rectangular matrix by matrices of smaller rank as the limit of lp-apprpoximation

In this paper we investigate a connection between l_p-approximation and the Chebyshev approximation of a rectangular matrix by matrices of smaller rank. We consider also the stationary points of problems (4) and (5) which are connected with these approximations.     

Numerical solution of differential equations using Sinc method based on the interpolation of the highest derivatives

In general, we will use the numerical differentiation when dealing with the differential equations. Thus the differential equations can be transformed into algebraic equations and then we can get the numerical solutions. But as we all have known, the numerical differentiation process is very sensitive to even a small level of errors. In contrast it is expected that on average the numerical integration process is much less sensitive to errors. In this paper, based on the Sinc method we provide a new method using Sinc method incorporated with the double exponential transformation based on the interpolation of the highest derivatives (SIHD) for the differential equations. The error in the approximation of the solution is shown to converge at an exponential rate. The numerical results show that compared with the exiting results, our method is of high accuracy, of good convergence with little computational efforts. It is easy to treat nonhomogeneous mixed boundary condition for our method, which is unlike the traditional Sinc method.     

Local polynomial reproduction and moving least squares approximation

Local polynomial reproduction is a key ingredient in providingerror estimates for several approximation methods. To boundthe Lebesgue constants is a hard task especially in a multivariatesetting. We provide a result which allows us to bound the Lebesgueconstants uniformly and independently of the space dimensionby oversampling. We get explicit and small bounds for the Lebesgueconstants. Moreover, we use these results to establish errorestimates for the moving least squares approximation scheme,also with special emphasis on the involved constants. We discussthe numerical treatment of the method and analyse its effort.Finally, we give large scale examples.     

二元二次函数逼近的存在性和局部性

This paper analysis the local behavior of the bivariate quadratic function approximation to a bivariate function which has a given power series expansion about the origin. It is shown that the bivariate quadratic Hermite-Pade form always defines a bivariate quadratic function and that this function is analytic in a neighborhood of the origin.     

Trapezoidal and triangular approximations preserving the expected interval

In this paper, an improvement of the nearest trapezoidal approximation operator preserving the expected interval is studied, which is proposed by Grzegorzewski and Mrówka. A formula for computing the improved approximation is presented. Moreover, the nearest triangular approximation operator preserving the expected interval is also investigated.     

由子基生成的内部算子和闭包算子

本文研究粗糙集与拓扑空间的关系,统一地使用拓扑空间中的集合关于子基的内部和闭包来研究粗糙集理论和覆盖广义粗糙集理论中的下近似集和上近似集,以及由它们导出的关于子基的开集,导集,闭集,边界．研究这两个概念及由它们导出的相关概念的性质不仅对于粗糙集理论,而且对于拓扑学本身都有重要的理论和实际应用意义．     

The strong approximation property and the weak bounded approximation property

We show that the strong approximation property (strong AP) (respectively, strong CAP) and the weak bounded approximation property (respectively, weak BCAP) are equivalent for every Banach space. This gives a negative answer to Oja's conjecture. As a consequence, we show that each of the spaces _c0$c_0$ and _?1${?}_1$ has a subspace which has the AP but fails to have the strong AP.     

On polynomial approximation in the uniform norm by the discrete least squares method

The discrete least squares method is convenient for computing polynomial approximations to functions. We investigate the possibility of using this method to obtain polynomial approximants good in the uniform norm, and find that for a given set ofm nodes, the degreen of the approximating polynomial should be selected so that there is a subset ofn+1 nodes which are close ton+1 Fejér points for the curve. Numerical examples are presented.Sponsored by the United States Army under Contract No. DAAG29-80-C-0041.     

Approximation algorithms for combinatorial fractional programming problems

We are concerned with a combinatorial optimization problem which has the ratio of two linear functions as the objective function. This type of problems can be solved by an algorithm that uses an auxiliary problem with a parametrized linear objective function. Because of its combinatorial nature, however, it is often difficult to solve the auxiliary problem exactly. In this paper, we propose an algorithm which assumes that the auxiliary problems are solved only approximately, and prove that it gives an approximate solution to the original problem, of which the accuracy is at least as good as that of approximate solutions to the auxiliary problems. It is also shown that the time complexity is bounded by the square of the computation time of the approximate algorithm for the auxiliary problem. As an example of the proposed algorithm, we present a fully polynomial time approximation scheme for the fractional 0–1 knapsack problem.     

Saddlepoint Approximation for the Distribution Function Near the Mean

We repair numerical difficulties in applying saddlepoint tail probability approximations when the ordinate at which the approximation is evaluated is near the mean of the distribution approximated. These modifications apply to double saddlepoint approximations to conditional distributions as well.     

Extreme value theory for dependent sequences via the stein-chen method of poisson approximation

In 1970 Stein introduced a new method for bounding the approximation error in central limit theory for dependent variables. This was subsequently developed by Chen for Poisson approximation and has proved very successful in the areas to which it has been applied. Here we show how the method can be applied to extreme value theory for dependent sequences, focussing particularly on the nonstationary case. The method gives new and shorter proofs of some known results, with explicit bounds for the approximation error.     

A -functional and the rate of convergence of some linear polynomial operators

We show that the -functional

where , is equivalent to the rate of convergence of a certain linear polynomial operator. This operator stems from a Riesz-type summability process of expansion by Legendre polynomials. We use the operator above to obtain a linear polynomial approximation operator with a rate comparable to that of the best polynomial approximation.

     

Weighted Best Local Approximation

In this survey, the notion of a balanced best multipoint local approximation is fully exposed since they were treated in the Lpspaces and recent results in Orlicz spaces. The notion of balanced point, which was introduced by Chui et al. in 1984 are extensively used.     

Can the Kerr Solution Be Found by the Einstein–Infeld–Hoffmann Method?

We present expansions of the Kerr metric in harmonic coordinates for the values of the radial coordinate r at which the dimensionless parameters m/r and a/r (m and a are parameters used in the Kerr solution) are of the respective second and first orders of smallness. We show that it is impossible to obtain these expansions uniquely using the Einstein–Infeld–Hoffmann method. We conclude that we must normalize the Kerr metric expansions for the expressions obtained in deriving the equations of translational motion of particle singularities and the evolution equations of their spins in the post-Newtonian and higher-order approximations.     

Lipschitz continuity and Gateaux differentiability of the best approximation operator in vector-valued Chebyshev approximation

When G is a finite-dimensional Haar subspace of C(X,R^k), the vector-valued functions (including complex-valued functions when k is 2) from a finite set X to Euclidean k-dimensional space, it is well-known that at any function f in C(X,R^k) the best approximation operator satisfies the strong unicity condition of order 2 and a Lipschitz (Hőlder) condition of order . This note shows that in fact the best approximation operator satisfies the usual Lipschitz condition of order 1 and has a Gateaux derivative on a dense set of functions in C(X,R^k).     

Jackson type theorems on complex curves

Let Γ be a closed smooth Jordan curve in the complex plane. In this paper, with the help of a class of fundamental functions of Hermite interpolation, the author introduces a continuous function interpolation which uniformly approximates to f(z) ε C(Γ) with the same order of approximation as that in Jackson Theorem 1 on real interval [−1, 1]. The accuracy of the order of approximation is proved. Using the method different from the early works, the author studies simultaneous approximation to function and its derivatives and the desired results analogues to that in Jackson Theorem 2 on real interval [−1, 1] are obtained.      

回归函数的有理逼近模型分析与研究

经济问题的研究中存在着大量的回归分析问题 ,但变量之间的关系往往是非线性的 .传统的建模原则往往对问题作了一系列的假设 ,因而模型不具有普遍适应性 .本文引进了一种特殊的非参数估计方法——回归函数的有理逼近 ,它与最小二乘法相比 ,提高了拟合与预测的精度 .     

On 3-monotone approximation by piecewise polynomials

We consider 3-monotone approximation by piecewise polynomials with prescribed knots. A general theorem is proved, which reduces the problem of 3-monotone uniform approximation of a 3-monotone function, to convex local L₁ approximation of the derivative of the function. As the corollary we obtain Jackson-type estimates on the degree of 3-monotone approximation by piecewise polynomials with prescribed knots. Such estimates are well known for monotone and convex approximation, and to the contrary, they in general are not valid for higher orders of monotonicity. Also we show that any convex piecewise polynomial can be modified to be, in addition, interpolatory, while still preserving the degree of the uniform approximation. Alternatively, we show that we may smooth the approximating piecewise polynomials to be twice continuously differentiable, while still being 3-monotone and still keeping the same degree of approximation.     

An application of the generalized alternating polynomials to the numerical solution of Fredholm integral equations

Generalized alternating polynomials have been introduced by the author earlier. In the present paper their indirect analogue is constructed for numerical solution of the Fredholm linear integral equations. Although the proposed method is a particular case of the general projection scheme, its valuable feature is the presence of a sequence of parameters, which, for sufficiently smooth kernels and inhomogeneous terms, serves as an indicator of the quality of approximation.     

Feedback approximation of the minimum energy control for linear infinite-dimensional systems with zero terminal state

The paper studies the minimum energy control problem for linear infinite-dimensional systems with an unbounded input operator and zero terminal state. This problem is approximated by the minimum energy control problem with a small terminal state for which the solution is derived in feedback form. The operators which comprise the feedback are described in terms of differential relations which, depending on circumstances, involve Liapunov or Riccati differential equations. A detailed example illustrates how the general results apply to the wave equation with control in Dirichlet boundary condition.This work was supported by the Polish Ministry of National Education under Grant DNS-T/02/097/90-2.