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一致收敛判别方法的探讨摘要一致收敛理论是数学分析的一个重要的研究分支.一致收敛概念及判定的掌握是学习数学分析的重点和难点,而且一致收敛在泛函分析、偏微分方程等学科中也有广泛而深入的应用.本文首先简单阐述函数列、函数项级数及含参量反常积分一致收敛的概念,然后从函数列、函数项级数及含参量反常积分三方面着手,分别列出常用的判别一致收敛的方法,并由常用的方法推出一些定理.本文在判别函数列一致收敛的方法探索中,由函数列的两边夹判别法推得一种比式判别法；并利用L条件,给出函数列一致L条件的定义,研究满足一致L条件的函数列的一致收敛性；研究在函数列可微条件下,它的导函数列在一致有界时,函数列的一致收敛性.在判别函数项级数一致收敛的方法探索中,给出函数项级数一致L条件的定义,研究满足一致L条件的函数项级数的一致收敛性.在文献[2]中一些未给出证明的定理,在本文中也将给出简单的证明.关键词:函数列；函数项级数；含参量反常积分；一致收敛InvestigateontheCriterionofUniformConvergenceMathematicsandAppliedMathematics2006-2JiangSu-pingSupervisorLiangZhi-qingAbstractUniformConvergencetheoryisanimportantresearchbranchofmathematicalanalysis.Theunderstandingandjudgingofthisconceptionarethekeyaswellasdifficultpointofmathematicalanalysis.Furthermore,UniformConvergencehasbeenwidelyusedinthesubjectsofFunctionalAnalysisandPartialDifferentialEquations.ThisarticlewillfirstbrieflyexplaintheFunctionColumn,SeriesofFunctionsandParameterImproperconceptofuniformconvergence.Then,outfromthreeaspects,namelythefunction,thefunctionparametersoftheSeriesandtheinfiniteintegrationwithparameter,itwilllistsomemethodscommonlyusedintheidentificationofUniformConvergencefromwhichsometheoremwillbededuced.IntheresearchofthemethodsofidentifyingUniformConvergence,anotherkindofidentifyingmethodcalledRatiomethodisdeducedthroughbetweendiscriminantmethod.Besides,takingadvantageofLcondition,thispaperwilldefineUniformLconditionanddiscussesConvergenceunderLcondition.Besides,itwilldiscussetheUniformConvergenceoffunctionwhenitsderivedfunctionsareuniformlyboundedundermicro-conditions.IntheresearchofthemethodsofidentifyingUniformConvergenceofSeries,thispaperwillgivethedefinitionofLconditionofUniformConvergenceofSeriesanddiscussesUniformConvergenceofSeriesunderLcondition.Theoremsthathasnotbeenprovedindocument2willalsobebrieflyprovedinthispaper.Keywords:functioncolumn;seriesoffunctions;infiniteintegrationwithparameter;uniformconvergence目录0前言·······································································································11预备知识·································································································22一致收敛的判别方法··················································································62.1函数列一致收敛的判别方法···································································62.1.1常用方法··················································································62.1.2两边夹判别法··········································································102.1.3单调判别法·············································································112.1.4一致L条件判别法····································································132.1.5导数判别法·············································································142.1.6点列判别法·············································································152.2函数项级数一致收敛的判别方法···························································162.2.1常用方法················································································162.2.2两边夹判别法··········································································202.2.3比较判别法·············································································212.2.4单调判别法·············································································222.2.5一致L条件判别法·····································································232.2.6导数判别法·············································································242.2.7点列判别法·············································································262.3含参量反常积分一致收敛的判别方法····················································272.3.1常用方法················································································272.3.2两边夹判别法··········································································292.3.3比较判别法·············································································292.3.4单调判别法·············································································312.3.5点列判别法·············································································31结束语·····································································································31致谢········································································································31参考文献··································································································3210前言一致收敛的理论是数学分析的重要组成部分之一,也是学好后继课程,如泛函分析、偏微分方程等的必备基础.一致收敛是数学分析教学中的难点之一,尤其是涉及到函数列、函数项级数与含参量反常积分的一致收敛性问题.数学分析中的积分运算与其它运算的可交换性,我们就需要探讨它们的一致收敛性来作为保证.目前,已有许多文献对一致收敛进行了研究.如在文献[1]中编者介绍了函数列、函数项级数及含参量反常积分一致收敛的概念,并介绍了判别函数列、函数项级数及含参量反常积分一致收敛的充要条件；文献[2]对一致收敛分别从定义、充要条件、一般性质、运算法则、判别方法等方面做了讨论；文献[3]给出了判别函数列一致收敛性的一种方法,这种方法与Dini定理的区别在于:Dini定理是数列单调,而作者所给的是函数单调.文献[4]介绍了函数项级数中的Dini定理.文献[5]则是对函数项级数的导数所需满足怎样的条件才能使级数一致收敛进行探讨,从而得到了函数项级数一致收敛的导数判别法.虽然已有诸多文献对一致收敛进行了研究,但多数只是就某单一方面进行研究.本文试图从函数列、函数项级数以及含参量反常积分一致收敛的判别方法进行探索.在文献[2]中未给出证明的定理,本文也将给出简单的证明.本文可分为两大部分,第一部分简单阐述函数列、函数项级数及含参量反常积分一致收敛的概念,同时给出函数列一致L条件及函数项级数一致L条件.第二部分是本文的主要内容,从函数列、函数项级数以及含参量反常积分三方面着手,分别列出常用的判别一致收敛的方法,并由常用的方法推出一些定理.