保凸性的代数多项式在L中的逼近

设f(x)∈L[-1,1].以∏_n表示阶不超过n的代数多项式的全体.我们已经熟知∏_n关于f(x)在L中的最佳逼近E_(f)_L可以用它的L中的k阶光滑模w_k(f,1/n)_L来刻划的事实:但是,当被逼近的函数f(x)是凸函数时,如果我们限制去逼近的代数多项式也是凸的,那么对于相应的逼近度能得到什么样的估计呢?以∏_n~*表示∏_n中的所有凸的多项式的全体.     

新题征展(32)

A　题组新编1.对于任意 x∈ R函数 f (x)都满足f(x 2 ) =f (2 - x) .(1)如果方程 f(x) =0恰好有 2 0 0 2个不同实根 ,则这些根之和为 (　 ) .(A) 0　 (B) 2 0 0 2　 (C) 4 0 0 4　 (D) 80 0 8(2 )如果方程 f (x 2 ) =0恰好有 2 0 0 2个不同的实根 ,则这些根之和为 (　 ) .(A) 0　 (B) 2 0 0 2　 (C) 4 0 0 4　 (D) 80 0 8(3)如果方程 f(x - 2 ) =0 ,恰好有 2 0 0 2个不同的实根 ,则这些根之和为 (　 ) .(A) 0　 (B) 2 0 0 2　 (C) 4 0 0 4　 (D) 80 0 8(4)如果方程 f (x 1) =0恰好有 2 0 0 3个不同的实根 ,则这些根之和为 (　 ) .(…     

几何凸函数的若干性质

1 预备知识 设函数f(x)在区间M上有定义,如果对于任意x0,x2∈M和t∈(0,1)都有 f[tx1+(1-t)x2]≤tf(x1)+(1-t)f(x2)(1)则说f(x)在M上为下凸的.如果(1)中的不等号反向,则说f(x)在M上为上凸的. 显然,如果f(x)在M上是上凸的,则-f(x)     

对不完整的不定积分的补充修正

如果在区间I上连续函数f(x)的原函数除有限个点外处处可导,则可通过补充定义这些点上的值使之“完整”起来,从而可以用它来求出f(x)在I任意闭子区间上的定积分.     

关于Jackson算子的最佳逼近常数

本文求出用Jackson算子J_n(f;x)逼近C_2n中的函数f(x)时的准确逼近常数:对?n≥1,有|J_n(f;x)-f(x)|≤(4-6/π)ω(f;1/n)及用阶数不超过n的三角多项式对函数f(x)的最佳逼近常数的上界估计:?n≤1,有K_n(f)_e≤(7-(21)/(2π))ω(f;1/n)     

试题研讨(18)

题1设f(x)2-x-a (x≤0),f(x-1) (x＞0).若f(x)=x有且仅有两个实数解.则实数a的取值范围是( ).     

爵克松奇异积分对连续函数逼近的准确常数

<正> 1.设 C_(2x)是周期2π的连续函数的全体,E_n(f)表示阶数不超过 n 的三角多项式对函数 f(x)的最佳逼近,ω(f;δ)表示函数 f(π)的连续模.(?)证明:如果f(x)∈C_(2π)的话,     

三角恒等式与三角方程

设有两个函数y=f1（x）与y=f2（x），如果对任意x0∈D都有f1（x0）=f2（x0），则称f1（x）=f2（x）是D上的恒等式，如果f1（x），f2（x）中有一个是三角函数式，就称此恒等式为三角恒等式。     

关于图是(g,f,n)-临界图的充分条件

设G是一个图. 设g和f是两个定义在V(G)上的整值函数使得对V(G)所有的顶点x有g(x)f(x). 图G被称为(g,f,n)-临界图,如果删去G的任意n个顶点后的子图都含有G的(g,f)-因子. 本文给出了图是(a,b,n)-临界图几个充分条件. 进一步指出这些条件是最佳的. 例如,如果对V(G)所有的顶点x和y都有g(x)＜f(x), n+g(x)dG(x)和g(x)/(dG(x)-n)f(y)/dG(y),则G是(g,f,n)-临界图.     

广义凸函数性质初探

设函数f（x）在区间I上有定义．如果对于任意x_1、x_2∈I和t∈（0，1）有则说f（x）在I上为下凸的．如果对于任意x_1、x_2∈I和t∈（0，1）有则说f（x）在I上为上凸的，如果对于一切t∈（0，1），（1）式（（2）式）中的等式仅当x_1＝x_2时成立，则说f（x）在I上为严格下凸（严格上凸）的．显然，如果f（x）在I上是上凸的，则－f（x）在I上就是下凸的．为此，以下我们着重讨论下凸函数由［1］，若函数f（x）在区间I上连续且对任意x_1、x_2∈I成立，则f（x）在I上是下凸的．对于凸函数，我们有著名的Jensen不等式：命题1设f（x）是区间…     

Cournot tâtonnement and dominance solvability in finite games

If a finite strategic game is strictly dominance solvable, then every simultaneous best response adjustment path, as well as every non-discriminatory individual best response improvement path, ends at a Nash equilibrium after a finite number of steps. If a game is weakly dominance solvable, then every strategy profile can be connected to a Nash equilibrium with a simultaneous best response path and with an individual best response path (if there are more than two players, switches from one best response to another may be needed). Both statements remain valid if dominance solvability in the usual sense is replaced with “BR-dominance solvability”, where a strategy can be eliminated if it is not among the best responses to anything, or if it is not indispensable for providing the best responses to all contingencies. For a two person game, some implications in the opposite direction are obtained.     

关于Hayman问题与分担值

设 f 为非常数亚纯函数, n(≥ 4)为正整数, a 为非零有穷复数. 若a 为f ⁿ与 (f ⁿ)′ 的CM分担值, 则nf ’=f.     

关于Hayman问题与分担值

该文主要讨论Hayman的一个问题与分担值的联系，并运用新颖的方法证明了：设f 为非常数的整函数，n,k均为正整数，n≥k+1,a为非零的有穷复数，若a为f^n与(f^n)^(k)的分担值，则nf′=ωf，ω为方程t^k=1的根.     

关于有界全纯函数在边界附近的某些性质

<正> 在[1]中 H.G.Eggleston 曾经证明了如下一个很有用的定理.设 f(z)是区域 D 内的有界全纯函数并 z_0为 D 的某一界点,z_0可为∞,但 D 至少人含有一有限还点为其界点.让 L 是一弧而以 z_0为其一端点且其他各点全属 D 内.若     

The continuity of the metric projection on a subspace of finite codimension in the space of continuous functions

The closed subspaces of finite codimension of the space C(X) of all continuous real-valued functions on a compact Hausdorff space X, for which the set of elements of best approximations of every function f C(X) is nonempty and compact, are characterized. It is shown that if the compact Hausdorff space X is infinite, then C(X) has no subspace of a finite Codimension n > 1 which has a nonempty set of elements of the best approximation for an arbitrary function f 6 (X) and which has an upper-semicontinuous metric projection.Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 531–539, April, 1976.     

Montel正规定则的推广(英文)

用简单的方法证明了全纯函数族的一个正规定则,推广了Montel正规定则.设F为区域D上的一个全纯函数族,其零点是重级的,a为有限复数.如果f,g∈F,有(f-1)f′与(g-1)g′分担a,则F正规.     

Constrained optimization involving expensive function evaluations: A sequential approach

This paper presents a new sequential method for constrained nonlinear optimization problems. The principal characteristics of these problems are very time consuming function evaluations and the absence of derivative information. Such problems are common in design optimization, where time consuming function evaluations are carried out by simulation tools (e.g., FEM, CFD). Classical optimization methods, based on derivatives, are not applicable because often derivative information is not available and is too expensive to approximate through finite differencing.The algorithm first creates an experimental design. In the design points the underlying functions are evaluated. Local linear approximations of the real model are obtained with help of weighted regression techniques. The approximating model is then optimized within a trust region to find the best feasible objective improving point. This trust region moves along the most promising direction, which is determined on the basis of the evaluated objective values and constraint violations combined in a filter criterion. If the geometry of the points that determine the local approximations becomes bad, i.e. the points are located in such a way that they result in a bad approximation of the actual model, then we evaluate a geometry improving instead of an objective improving point. In each iteration a new local linear approximation is built, and either a new point is evaluated (objective or geometry improving) or the trust region is decreased. Convergence of the algorithm is guided by the size of this trust region. The focus of the approach is on getting good solutions with a limited number of function evaluations.     

Functions with prescribed best linear approximations

A common problem in applied mathematics is that of finding a function in a Hilbert space with prescribed best approximations from a finite number of closed vector subspaces. In the present paper we study the question of the existence of solutions to such problems. A finite family of subspaces is said to satisfy the Inverse Best Approximation Property (IBAP) if there exists a point that admits any selection of points from these subspaces as best approximations. We provide various characterizations of the IBAP in terms of the geometry of the subspaces. Connections between the IBAP and the linear convergence rate of the periodic projection algorithm for solving the underlying affine feasibility problem are also established. The results are applied to investigate problems in harmonic analysis, integral equations, signal theory, and wavelet frames.     

新组合型的三角插值多项式

将被插函数进行组合平均,构造一个新组合型的三角插值多项式Cn(f;t,x),使得它在全轴上一致收敛到每个以2π为周期的连续函数,且对Cj2π连续函数类的逼近阶达到最佳,这里0jt,t为任给的奇自然数.     

涉及导函数的亚纯函数的唯一性

本文研究亚纯函数f与f＇具有两个公共值时的Frank问题,证明了若f与f＇以0为CM公共值,以有穷非零复数b为IM公共值,则f=f＇或f=2b/1-ce-2z,其中c为任一有穷非零复数.