关于q-Bernstein多项式的加速问题

In this paper,we investigate not only the acceleration problem of the q-Bernstein polynomials Bn(f,q;x)to B∞(f,q;x)but also the convergence of their iterated Boolean sum.Using the methods of exact estimate and theories of modulus of smoothness,we get the respective estimates of the convergence rate,which suggest that q-Bernstein polynomials have the similar answer with the classical Bernstein polynomials to these two problems.     

ω,q-Bernstein多项式的Voronovskaya-型公式

讨论了ω,q-Bernstein多项式的Voronovskaya-型公式及其收敛的饱和性.给出了当01[0,1]时ω,q-Bernstein多项式的Voronovskaya-型公式.如果0<ω,q<1,f∈C1[0,1]时ω,q-Bernstein多项式的Voronovskaya-型公式.如果0<ω,q<1,f∈C¹[0,1],则ω,q-Bernstein多项式的收敛阶为o(q1[0,1],则ω,q-Bernstein多项式的收敛阶为o(qⁿ)当且仅当((f(1-qn)当且仅当((f(1-q^(k-1)-f(1-q)(k-1)-f(1-q)^k))/((1-qk))/((1-q^(k-1)-(1-q(k-1)-(1-q^k)))=f'(1-qk)))=f'(1-q^k),k=1,2,…还证明f如果f在[0,1]是凸的或者在(-ε,1+ε)(ε>0)解析,则ω,q-Bernstein多项式的收敛阶为o(qk),k=1,2,…还证明f如果f在[0,1]是凸的或者在(-ε,1+ε)(ε>0)解析,则ω,q-Bernstein多项式的收敛阶为o(qⁿ)当且仅当f是线性函数.     

CONVERGENCE RATE FOR ITERATES OF q-BERNSTEIN POLYNOMIALS

Recently, q-Bernstein polynomials have been intensively investigated by a number of authors. Their results show that for q ≠ 1, q-Bernstein polynomials possess of many interesting properties. In this paper, the convergence rate for iterates of both q-Bernstein polynomials and their Boolean sum are estimated. Moreover, the saturation of {Bn（., qn）} when n → ∞ and convergence rate of Bn（f,q;x） when f ∈ C^n-1 [0, 1], q → ∞ are also presented.     

On Polarized Surfaces (X, L) with h0(L) > 0, κ(X) ≥ 0 and g(L) = q(X) + 1

Let (X,L) be a polarized surface. If h⁰(L)>0, then g(L) q(X). In our previous papers, we classified polarized surfaces (X,L) with g(L)=q(X) and h⁰(L)>0. In this paper, we classify polarized surfaces (X,L) with g(L)=q(X)+1, h⁰(L)>0, and (X) 0.     

反常积分敛散性的根值判别法

由于积分与级数在理论上是统一的,因此有关正项级数的根式判别法可被推广以判别无穷限积分和瑕积分的敛散性.设f(x)是[a,+∞)上的非负函数,li mx→+∞xf(x)=ρ,则当ρ1时,反常积分∫a+∞f(x)dx收敛,而当ρ1时,反常积分∫a+∞f(x)dx发散;设f(x)是(a,b]上的非负函数,a为瑕点,xli→ma+(f(x))x-a=ρ,则当ρ1时,反常积分∫abf(x)dx收敛,而当ρ1时,反常积分∫baf(x)dx发散.     

二阶差分方程解的振动性与渐近性

本文研究了二阶差分方程Δ(a     

在零点的隣區內彼此相等的特徵函数

<正> §1.引言 大家知道,兩個不相恆等的特徵函數(以下简称特函)可以在零點的隣區內相等。為固定用語起見,在本文中我們說特函f(t)属於集合(U),如果存在一個特函,它与f(t)在零的隣區內相等,但並不恆等於f(t);如果f(t)不屬於(U),就說它屬於(U)。     

Lagrange插值逼近导数的平均收敛

<正>We consider the rate of mean convergence of derivatives by Lagrange interpolation operators L_n(f,x) based on the zeros of Chebyshev polynomials of the first kind.A sharp estimate of the derivative of L_n(f,x)—f(x) in terms of the error of best approximation by polynomials of degree n is derived.     

用黎曼求和法(R,1)求和時的基卜斯現象

<正> 假設級數滿足下面兩個條件,即:(甲)在原點的某一鄰域內,對於h(≠0)的一切值級數收斂;     

关于二阶完全非线性抛物组的差分格式

则称二阶完全非线性组(1)是一致抛物的.我们在矩形域Q_T={0≤x≤l,0≤t≤T}(l>0,T>0)上研究方程组(1)满足边界条件     

无穷限反常积分收敛时被积函数的极限状态

根据无穷限反常积分∫a^＋∞f（x）dx收敛的柯西准则和定积分的性质,讨论被积函数f（x）当x→∞时。的极限状态,并得出当无穷限反常积分∫a^＋∞f（x）dx收敛且f（x）在[a,＋∞）上连续,或者无穷限反常积分∫a^＋∞f（x）dx绝对收敛时,存在数列{xn}∩[a,＋∞]且xn→＋∞（n→∞）,使limn→∞xnf（xn）=0.     

AN UPBOUND OF HAUSDORFF'S DIMENSION OF THE DIVERGENCE SET OF THE FRACTIONAL SCHR(O)DINGER OPERATOR ON Hs(Rn)

Given n ≥ 2 and a ＞ 1/2,we obtained an improved upbound of Hausdorff's dimension of the fractional Schr(o)dinger operator; that is,sup f∈Hs(Rn) dimH{x∈Rn:limt→0...     

A selection lemma for sequences of measurable sets,and lower semicontinuity of multiple integrals

In the present paper we show that the integral functional is lower semicontinuous with respect to the joint convergence of y_k to y in measure and the weak convergence of u_k to u in L₁. The integrand f: G × ^N × ^m , (x, z, p) f(x, z, p) is assumed to be measurable in x for all (z,p), continuous in z for almost all x and all p, convex in p for all (x,z), and to satisfy the condition f(x,z,p)(x) for all (x,z,p), where is some L₁-function.The crucial idea of our paper is contained in the following simple     

基于凸优函数的Newton类方法的收敛定理

本文研究了求解Banach空间上非线性算子方程f(x)=0的Newton类方法的收敛性.利用优函数原理,在A(x0)1f满足关于某一凸优函数的广义Lipschitz条件下,得到了Newton类方法的一个半局部收敛定理.同时,当f和A(x)及初始点x0给定时,针对广义Lipschitz条件构造了相应的优函数,推广了Newton类方法的相关结果.     

多线性分数次积分算子一般框架下的交换子的有界性——献给陆善镇教授75华诞

对任意给定的正整数m,Z^＋×{1,...,m}的任意一个有限子集S,定义一般化的多线性分数次积分算子的交换子Iα,→b,S（f）（x）=∫（Rn）^m ∏（i,j）∈S（bi（x）-bi（yj））/（｜x-y1｜＋…＋｜x-ym｜）^mn-α∏（j=1→m）fj（yj）d→y,其中d→y=dy1…dym.此框架下的交换子包含了以往研究的各类分数次积分算子的交换子,并蕴含了多线性背景下新的交换子形式.在上述非常一般框架下,本文给出带多重A→p,q权的多线性分数次积分算子的交换子Iα,→b,S（→f）的加权强型（L^p1（ω1）×···×L^pm（ωm）,L^q（ν→ωq））估计和加权弱型端点估计.本文还得到更一般核条件下的上述结果.     

积分中值定理中间点比较及有关平均不等式

中值定理中间点是区间端点的平均.设f (x)、g(x)在同一区间[a,b]内严格单调并可积,p(x)、q(x)恒正可积,按积分中值定理各有唯一的中间点ξf ,p(a,b)和ξg,q(a,b) .当f递增(减)且f (g- 1)凸(凹)时,有ξg,p(a,b) <ξf,p(a,b) ;当p(x)q(x) 递增(减)且q(x) ∫bap(x) dx >( <) 0时,有ξf,q(a,b) <ξf ,p(a,b) .由此可证明和发现一系列有关平均的不等式.     

在一个新步长规则下梯度投影算法的全局收敛性

考虑约束最优化问题：minx∈Ωf(x)其中:f:R^n→R是连续可微函数，Ω是一闭凸集。本文研究了解决此问题的梯度投影方法，在步长的选取时采用了一种新的策略，在较弱的条件下，证明了梯度投影响方法的全局收敛性。     

On Convergence of Minimization Methods: Attraction, Repulsion, and Selection

In this paper, we revisit the convergence properties of the iterationprocess x^k+1=x^k–(x^k)B(x^k)^–1f(x^k)for minimizing a function f(x). After reviewing some classic results andintroducing the notion of strong attraction, we give necessary andsufficient conditions for a stationary point of f(x) to be a point of strongattraction for the iteration process. Not only this result gives a newalgorithmic interpretation to the classic Ostrowski theorem, but alsoprovides insight into the interesting phenomenon called selectiveminimization. We present also illustrative numerical examples for nonlinearleast squares problems.     

凸函数的双参数平均不等式的新证明

利用 Tchebycheff积分不等式和积分形式的 Cauchy中值定理证明了下列结论 :设 f(x)是 [a,b]上的正连续函数 ,且在 (a,b)内可微 ,若 f′(x)单调递增 ,则对任意的 p,q,有 Mp,q(f) 相似文献   

线段映射的局部变差增长与局部拓扑熵

本文考虑闭区间上变差有界的连续映射f:I→I的局部变差增长γ(x,f)与局部拓扑熵h(x,f)．将证明γ(x,f)≥h(x,f)对所有x∈I成立,并且局部变差增长映射γf(x)=γ(x,f)与局部拓扑熵映射sf(x)=h(x,f)都是上半连续的,得到一个变分原理:局部变差增长γ(x,f)与局部拓扑熵h(x,f)的上确界分别等于全局变差增长γ(f)=limn→∞1/nln Var(fn)与拓扑熵h(f)．当映射f:I→I拓扑传递时,与Brin 和Katok对局部(测度)熵的讨论类似,我们证明,至多除一个不动点外,局部变差增长γ(x,f)与局部拓扑熵h(x,f)在开区间I°内恒为常值．