关于z.Ditzian定理

1987年Z.Ditzian提出了反映Bernstein算子收敛阶与所逼近函数光滑模之间关系的一个定理,并在α+β≤2情形下给出了这个定理的证明.对于α+β》2情形,Z.Ditzian给出了猜想.1992年周定轩证明了Z.Ditzian的猜想,完成了Z.Ditzian定理的证明.本文对于Z.Ditzian定理给出了一个新的直接证明,这个证明不需要讨论α,β的情况,而且还将Z.Ditzian定理拓广到Bernstein算子线性组合上.     

Bernstein算子线性组合的点态逆定理

本文给出了关于Bernstein算子线性组合的点态正逆定理,拓广了Z．Ditzian的两个结果．     

关于Bernstein多项式的一个注记

本给出了Z.Ditzian定理的一个简短证明。     

在C（S）空间中多元的Bernstein—Kantorovich算子的正逆定理

在C（S）空间中，对定义在单形上的Bernstein－Kantorovich算子Kn（f），给出了一个积分型估计式及一个弱型逆定理，得到了当0＜α＜1时，其中为定义在C（S）空间中的Ditzian－Totick光滑模[1]．     

关于Bernstein算子的收敛速度

本文对一类函数建立了Bernstein算子的一致逼近定理,而且给出了其逆定理的一个简短证明。     

Bernstein算子的强逆不等式

本文对Bernstein算子证明了其强逆不等式,这些不等式曾被Ditzian,Ivanov,Totik,李松等人用不同的方法得到过,但其结果是通常的估计(λ=1),古典的结果(λ=0)没有包含,本文引入κ-泛函K_λ~α(f,t~2)(0≤λ≤1,0<α<2),将已有结果推广到0≤λ≤1的情形。     

de Rham－Hodge－Signature算子的局部指标定理

整体的Ａｔｉｙａｈ－Ｓｉｎｇｅｒ指标定理对于一般的椭圆微分算子都成立．但对于局部指标定理来说，人们只能对具体的算子给予证明．并且各种情况需个别处理．由［６］知本文中证得的ｄｅＲｈａｍ－Ｈｏｄｇｅ－Ｓｉｇｎａｔｕｒｅ算子的局部指标既不是α型也不是β型的．这和经典的椭圆算子情形不同．     

de Rham－Hodge－Signature算子的局部指标定理

整体的Ａｔｉｙａｈ－Ｓｉｎｇｅｒ指标定理对于一般的椭圆微分算子都成立．但对于局部指标定理来说，人们只能对具体的算子给予证明．并且各种情况需个别处理．由［６］知本文中证得的ｄｅＲｈａｍ－Ｈｏｄｇｅ－Ｓｉｇｎａｔｕｒｅ算子的局部指标既不是α型也不是β型的．这和经典的椭圆算子情形不同．     

Bernstein 算子的同时逼近

借助于D itzian-T otik光滑模研究了Bernstein算子的同时逼近问题,给出了Bernstein算子同时逼近的正定理和等价定理.     

线性系统李代数 Segal-Shale-Weil 表示及全体 Kalman-Bucy 滤波器

设 ls_n 是具有|α|+|β|≤2性质的全体微分算子∑C_(αβ)~x~α ((?)β)/((?)x~β),C_(αβ)∈R,|α|=α_1+…+α_n.用 R~N 上的向量场,N=1/2(n~2+3n),本文构造了这个2n~2+3n+1维的李代数的一个表示.利用 Duncan-Mortensen-Zakai 方程及一个反同态,这个表示就给出了全体 n 维线性系统的 Kalman-Bucy 滤波器,换句话说,全体 Kalman-Bucy 滤波器合在一起定义了一个反同态 l_(sn)→V(R~N).一般说,这又断定 Kalman-Bucy 滤波器给出了“估计李代数”对向量场李代数的一个(反)同态.最后,ls_n 表示和 Segal-Shale-Weil 表示有密切的联系.     

RECENT PROGRESS ON SPHERICAL HARMONIC APPROXIMATION MADE BY BNU RESEARCH GROUP ——In memory of Professor Sun Yongsheng

As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the research on spherical harmonic analysis and approximation. In this paper, we incompletely introduce the main achievements in this area obtained by our group and relative researchers during recent 5 years （2001-2005）. The main topics are： convergence of Cesaro summability, a.e. and strong summability of Fourier-Laplace series; smoothness and K-functionals; Kolmogorov and linear widths.     

A unified approach to certain problems of best local approximation

In this paper we study best local quasi-rational approximation and best local approximation from finite dimensional subspaces of vectorial functions of several variables. Our approach extends and unifies several problems concerning best local multi-point approximation in different norms.     

Boundedness for commutators of multipliers on Hardy type spaces

In this paper, we study the commutators generalized by multipliers and a BMO function. Under some assumptions, we establish its boundedness properties from certain atomic Hardy space Hb^p（R^n） into the Lebesgue space L^p with p 〈 1.     

THE 8~(th) INTERNATIONAL CONGRESS ON INDUSTRIAL AND APPLIED MATHEMATICS

<正>August 10-14,2015Beijing,ChinaThe International Congress on Industrial and Applied Mathematics(ICIAM)is the premier international congress in the field of applied mathematics held every four years under the auspices of the International Council for Industrial and Applied Mathematics.From August 10 to 14,2015,mathematicians,scientists     

Towards Excellence in Science Chinese Academy of Sciences

<正>May 26,2014,Beijing Science is a human enterprise in the pursuit of knowledge.The scientific revolution that occurred in the 17th Century initiated the advances of modern science.The scientific knowledge system created by     

Bounds for the zeros of polynomials

Let P(z)=∑↓j=0↑n ajx^j be a polynomial of degree n. In this paper we prove a more general result which interalia improves upon the bounds of a class of polynomials. We also prove a result which includes some extensions and generalizations of Enestrǒm-Kakeya theorem.     

Boundedness of commutators related to Marcinkiewicz integrals on hardy type spaces

In this paper, the authors study the boundedness of the operator [μΩ, b], the commutator generated by a function b ∈ Lipβ(Rn)(0 ＜β≤ 1) and the Marcinkiewicz integrals μΩ, on the classical Hardy spaces and the Herz-type Hardy spaces in the case Ω∈ Lipα(Sn-1)(0 ＜α≤ 1).     

Jacobi polynomials used to invert the laplace transform

Given the Laplace transform F(s) of a function f(t), we develop a new algorithm to find an approximation to f(t) by the use of the classical Jacobi polynomials. The main contribution of our work is the development of a new and very effective method to determine the coefficients in the finite series expansion that approximation f(t) in terms of Jacobi polynomials. Some numerical examples are illustrated.     

Generalization of the interaction between Haar approximation and polynomial operators to higher order methods

In applications it is useful to compute the local average empirical statistics on u. A very simple relation exists when of a function f（u） of an input u from the local averages are given by a Haar approximation. The question is to know if it holds for higher order approximation methods. To do so, it is necessary to use approximate product operators defined over linear approximation spaces. These products are characterized by a Strang and Fix like condition. An explicit construction of these product operators is exhibited for piecewise polynomial functions, using Hermite interpolation. The averaging relation which holds for the Haar approximation is then recovered when the product is defined by a two point Hermite interpolation.