Simultaneous Approximation by Polynomial Projection Operators

Pointwise estimates are obtained for simultaneous approximation of a function f and its derivatives by means of an arbitrary sequence of bounded projection operators with some extra condition (1.3) (we do not require the operators to be linear) which map C_[-1,1] into polynomials of degree n, augmented by the interpolation of f at some points near ±1. The present result essentially improved those in [BaKi3], and several applications are discussed in Section 4.     

Approximation of discrete functions and Chebyshev polynomials orthogonal on the uniform grid

Let _N+2m ={−m, −m+1, …, −1, 0, 1, …,N−1,N, …,N−1+m}. The present paper is devoted to the approximation of discrete functions of the formf : _N+2m → ℝ by algebraic polynomials on the grid Ω _N ={0, 1, …,N−1}. On the basis of two systems of Chebyshev polynomials orthogonal on the sets Ω _N+m and Ω _N , respectively, we construct a linear operatorY _{n+2m, N} =Y _{n+2m, N} (f), acting in the space of discrete functions as an algebraic polynomial of degree at mostn+2m for which the following estimate holds (x ε Ω _N ):

超收敛中的双P猜想续篇——离散格林函数的权模估计

探讨超收敛猜想中p=4的情形.为此目的我们推导了离散格林函数的权模估计.     

球面上径向基函数最小范数插值逼近的误差估计

研究了球面径向基插值对球面函数的逼近问题,给出了一致逼近的上界估计式．文中结果说明,球面径向基插值的逼近阶会随函数光滑性的提高而增加．     

赋Orlicz范数的Orlicz空间中最佳逼近算子

在赋Ｏｒｌｉｃｚ范数的Ｏｒｌｉｃｚ空间中,给出最佳逼近算子单调性的一个充分条件和最佳逼近元存在定理．     

ON THE GENERALIZATIONS OF KALANDIA''''S LEMMA

Based on Bernstein's Theorem, Kalandia's Lemma describes the error estimate and the smoothness of the remainder under the second part of Holder norm when a Holder function is approximated by its best polynomial approximation. In this paper, Kalandia's Lemma is generalized to the cases that the best polynomial is replaced by one of its four kinds of Chebyshev polynomial expansions, the error estimates of the remainder are given out under Holder norm or the weighted Holder norms.     

ON NEWMAN-TYPE RATIONAL INTERPOLATION TO ｜x｜ AT THE CHEBYSHEV NODES OF THE SECOND KIND

Recently Brutman and Passow considered Newman-type rational interpolation to ｜x｜ induced by arbitrary set of symmetric nodes in [-1,1] and gave the general estimation of the approximation error.By their methods one could establish the exact order of approximation for some special nodes. In the present paper we consider the special case where the interpolation nodes are the zeros of the Chebyshev polynomial of the second kind and prove that in this case the exact order of approximation is O（1/n｜nn）     

Weighted inequalities and applications to best local approximation in Luxemburg norm

In this paper we establish a result about uniformly equivalent norms and the convergence of best approximant pairs on the unitary ball for a family of weighted Luxemburg norms with normalized weight functions depending on ε, when ε→ 0. It is introduced a general concept of Pade approximant and we study its relation with the best local quasi-rational approximant. We characterize the limit of the error for polynomial approximation. We also obtain a new condition over a weight function in order to obtain inequalities in Lp norm, which play an important role in problems of weighted best local Lp approximation in several variables.     

Approximation by <Emphasis Type="Italic">p</Emphasis>-Faber-Laurent Rational Functions in the Weighted Lebesgue Spaces

Let L C be a regular Jordan curve. In this work, the approximation properties of the p-Faber-Laurent rational series expansions in the weighted Lebesgue spaces L ^p(L, ) are studied. Under some restrictive conditions upon the weight functions the degree of this approximation by a kth integral modulus of continuity in L ^p(L, ) spaces is estimated.     

An Estimate for the <Emphasis Type="Italic">n</Emphasis>th Minimal Error of Linear Algorithms for One Problem in a Normed Vector Space

We find an estimate for the nth minimal error of linear algorithms for some problem defined in a finite-dimensional space with values in an arbitrary normed vector space.Original Russian Text Copyright © 2005 Sidorov S. P.The author was supported by the Russian Foundation for Basic Research (Grant 04-01-00060), the State Maintenance Programs for the Leading Scientific Schools of the Russian Federation (Grant 1295.2003.1), and the Program Universities of Russia (Grant 04.01.374).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 673–678, May–June, 2005.