Riemann积分的历史与发展

总结Riemann积分的发展历史、基本思想及其各种推广,介绍Riemann积分的局限性和缺陷,以及由此推动产生Lebesgue积分的过程.     

Lebesgue积分几何意义的推广

证明了非负有界函数的Lebesgue上积分等于函数下方图形的Lebesgue外测度,其Lebesgue下积分等于函数下方图形的Lebesgue内测度,从而将积分的几何意义从可测情形推广到一般情形.     

谈谈Lebesgue数钱

用Lebesgue关于数钱的比喻,解释Lebesgue积分的本质优点.     

瑕积分的收敛性与Lebesgue可积性

瑕积分的收敛性与Lebesgue可积性之间具有密不可分的关系.首先详细证明了瑕积分收敛定理,然后说明无穷积分的收敛定理与瑕积分的收敛定理是等价的,最后利用两条定理的等价性给出一个应用.     

用泛函观点看LEBESGUE积分

从泛函分析观点来看Lebesgue积分,使得Lebesgue积分可以用泛函分析最简单最基本的方法独立导出．基本做法是将Riemann对于区间[0,1]上的连续函数的积分看成连续函数空间C[0,1]上的连续线性泛函,再将它“自然”延拓到C[0,1]在积分范数意义下的完备化空间,而这个完备化空间正是Lebesgue可积函数空间L1[0,1]．     

多元Riemann可积函数的特征与完备化

给出了多元Riemann可积函数的基本特征,证明了多元Riemann可积函数空间的完备化是Lebesgue积分空间.     

变指标Lebesgue空间上的Marcinkiewicz积分高阶交换子

证明了一类Marcinkiewicz积分高阶交换子在变指标Lebesgue空间上的BMO和Lipschitz估计,对于分数次积分交换子也得到了类似的结果.     

多重奇异积分的多线性交换子

引进了由多重奇异积分和BMO函数生成的多线性交换子,然后得到了此类算子从Lebesgue积空间到Lebesgue空间的有界性,最后也给出了此类算子的加权和向量值不等式.     

Lebesgue积分在高等数学中的几个应用实例

实例说明Lebesgue积分在高等数学中求极限方面的几个实际应用．     

关于瑕积分与Lebesgue积分之关系的一种证法

本给出了函数f(x)的瑕积分绝对收敛时．必定Lebesgue可积的一种证明方法．     

The hu-meyer formula for non deterministic kernels

In this paper we prove the analogue of the Hu-Meyer formula for random kernels. More precisely, using a suitable notion of trace we give the relation between the multiple Stratonovich integral of a non adapted process and the multiple Skorohod integral     

Integral Representation with Adapted Continuous Integrand with Respect to Fractional Brownian Motion

We show that if a random variable is a final value of an adapted Hölder continuous process, then it can be represented as a stochastic integral with respect to fractional Brownian motion, and the integrand is an adapted process, continuous up to the final point.     

含控制参数的抽象动力方程连值问题之积分算子求解法

讨论一类抽象Volterra型积分算子,利用此获得含控制参数的抽象动力方程边值问题的解。这种新的求解法我们称为积分算子求解法。     

The anticipative Stratonovich integral in conuclear spaces

We define an anticipative stochastic integral of Stratonovich type with respect to a nonhomogeneous Wiener process in the dual of a nuclear space and investigate its basic properties.This research was partially supported by Komitet Bada Naukowych, Grant 2 1094 91 01.     

Almost every sequence integrates

The purpose of this paper is to discuss a first-return integration process which yields the Lebesgue integral of a bounded measurable function f: I → R defined on a compact interval I. The process itself, which has a Riemann flavor, uses the given function f and a sequence of partitions whose norms tend to 0. The “first-return” of a given sequence is used to tag the intervals from the partitions. The main result of the paper is that under rather general circumstances this first return integration process yields the Lebesgue integral of the given function f for almost every sequence . This research was initiated while the authors were in residence at the Mathematical Institute of St. Andrews University.     

Skorohod integral of a product of two stochastic processes

In this note we provide sufficient conditions for the product of a predictable and a backward predictable process to be Skorohod integrable.Supported in part by NSF No. INT-9401109, NSA No. MDR-90494H2094, and ONR No. N-00014-96-1-0262.     

流形上半鞅线积分与半鞅互变差的注记

给出了两流形上半鞅的互变差。     

Stochastic integration without tears

Stochastic integration theory is developed by axiomatizing the concept of semi-martingale in terms of a continuity property of integrals of simple functions. Using this approach, stochastic integration for left-continuous integrands, the change of variables formula and properties of the quadratic variation process are established in an elementary way. Submartingale decomposition theorems are introduced at a late stage in order to extend the results to general predictable integrands.     

关于Rosenblatt过程Wiener积分的随机Fubini定理(英文)

本文得到了一个关于Rosenblatt过程Wiener积分的随机Fubini定理,它可以视为经典Fubini定理的推广.