函数一致连续性的判别新法

分别给出了各类区间上函数一致连续性的新的判定方法,使得一些复杂函数的一致连续性可以借助于较易判别出一致连续性的初等函数去判定.最后举例验证了方法的可行性和有效性.     

函数一致连续的一个充要条件

给出了一元函数在区间上一致连续的一个充分必要条件,举例说明了使用它来讨论函数在区间上的一致连续性将更为简单.     

Fuzzy区间值函数的含参量积分及一致收敛性

在一元实函数无穷积分定义的基础上,定义了含参量Fuzzy区间值函数的正常积分和无穷积分,给出了含参量无穷积分一致收敛的定义和判定定理.     

一类非负函数无穷积分收敛性的刻画

针对无穷积分教学的需要,在利用随机理论的重要概念——失效函数来刻画出非负函数的递减趋于坐标横轴程度的基础上,运用一般推理方法及无穷积分敛散性的比较判别法,得到了一类非负函数的无穷积分敛散性的较为简便而有效的判别方法.     

无穷区间上两类抽象无界函数族相对紧的判定及其应用

给出了无穷区间上抽象无界连续函数族和抽象无界可微函数族相对紧性判定的充要条件,并应用它获得了无穷区间上二阶微分方程两点边值问题无界解的存在性.     

Riemann函数可积性的一个初等证明

本文将证明Riemaun函数有无穷多个间断点,但它在有限区间[a,b]上定积分存在,且其在区间[a,b]上的积分值为0.     

函数振幅的性质与应用

函数振幅是微积分中最基本的概念之一，它的性态决定了函数许多解析性质．本文中扩充了函数振幅的定义．并利用它定义了函数振动性强弱的概念．作为这一概念的应用，建立起判定函数连续性、一致连续性、可积性、有界变差性的比较判别法，并给出了一些应用实例．     

函数列在不同区间上一致收敛性的研究

函数列的一致收敛性与所讨论的区间有关.在区间的子区间或不同的区间上,函数列的一致收敛性表现如何呢?通过几个命题和几个实例,并利用几何画板可以帮助我们辨别函数列在不同区间上的一致收敛性.     

用导数判别函数的一致连续性

本文主要给出了用导数判别函数在一般区间上一致连续的方法,并举例说明不可以建立关于一致连续的比较判别法．     

最小发散级数和最大收敛级数不存在理论及其论证

无穷级数是高等数学中的一部分重要内容.在判断无穷级数收敛或发散时,比较原理(比较审敛法或比较判别法)是一个有效的判别法,其基本原理是:     

Sharpness of estimates for the least uniform derivation from the class of functions with bounded second derivative

We study the uniform approximation of continuous functions on an interval by functions with bounded second derivative. We prove the sharpness of estimates for the least uniform deviation of a function in terms of its local deviations on uniform and nonuniform three-point grids.     

Almost periodic solutions in the PC-space for uncertain impulsive dynamical systems

This paper provides sufficient conditions for the existence of almost periodic solutions for an uncertain impulsive dynamical system. The investigation is carried out by utilizing the concept of uniformly positive definite matrix functions, Hamilton-Jacobi-Riccati inequality and piecewise continuous functions of the Lyapunov functions type.     

On norm dependent positive definite functions

In the first part of the paper we prove a decomposition theorem for positive definite functions (Theorem 2.3) generalizing a result of de Leeuw and Glicksberg. Using this theorem, we then show (Theorem 3.1) that certain norm dependent positive definite functions are automatically continuous at every point different from zero.     

Local uniform error estimates for spherical basis functions interpolation

This paper discusses local uniform error estimates for spherical basis functions (SBFs) interpolation, where error bounds for target functions are restricted on spherical cap. The discussion is first carried out in the native space associated with the smooth SBFs, which is generated by a strictly positive definite zonal kernel. Then, the smooth SBFs are embedded in a larger space that is generated by a less smooth kernel, and for the target functions outside the original native space, the local uniform error estimates are established. Finally, some numerical experiments are given to illustrate the theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.     

Slutsky定理的一个推广

本文应用紧集上的连续函数是一致连续的性质，推广并证明了著名的Slutsky定理。     

Function spaces and product topologies on powers of spaces

A study is made of two classes of product topologies on powers of spaces: the general box product topologies, and the general uniform product topologies. Some examples are given and some results are shown about the properties of these general product spaces. This is applied to show that certain spaces of continuous functions with the fine topology are homeomorphic to box product spaces, and certain spaces of continuous functions with the uniform topology are homeomorphic to uniform product spaces.     

An estimate for the least uniform deviation from the class of functions with bounded second derivative

We consider the problem of uniform approximation of a continuous function on an interval by means of the class of functions with bounded second derivative. We prove an estimate for the least uniform deviation of a function in terms of its local deviations on uniform and nonuniform three-point grids.     

Approximation of continuous functions by broken lines with constraints on the angles between adjacent segments

The problem of best uniform approximation of a function continuous on an interval by piecewise linear continuous functions with fixed nodes and constraints on the angle between adjacent segments is considered.     

Laplace-Beltrami differentiability of positive definite kernels on the sphere

This contribution gives results on the action of the Laplace-Beltrami derivative on sufficiently smooth kernels on the sphere, those defined by absolutely and uniformly expansions generated by a family of at least continuous functions. Among other things, the results show that convenient Laplace-Beltrami derivatives of positive definite kernels on the sphere are positive definite too. We also include similar results on the action of the Laplace-Beltrami derivative on condensed spherical harmonic expansions.