傅立叶分析导论

图书标准信息

• 作者 Elias、Rami Shakarchi 著
• 出版社 世界图书出版公司
• 出版时间 2012-12
• 版次 1
• ISBN 9787510040559
• 定价 59.00元
• 装帧 平装
• 开本 24开
• 纸张 胶版纸
• 页数 311页
• 正文语种 英语
• 原版书名 Fourier Analysis: An Introduction

【内容简介】

　　《傅里叶分析导论》由在国际上享有盛誉普林斯大林顿大学教授Stein撰写而成，是一部傅立叶分析的入门教材，理论与实践并重，为了便于非数专业的学生学习，全书内容简明、易懂.全书分为三部分，第一部分介绍傅立叶级数的基本理论及其在等周不等式和等分布中的应用；第二部分研究傅立叶变换及其在经典偏微分方程及Radom变换中的应用；第三部分研究有限阿贝尔群上的傅立叶分析。书中各章均有练习题及思考题。目次：傅立叶积分的起源；傅立叶级数和基本性质；傅立叶级数的收敛性；傅立叶积分的应用；IR上的傅立叶变换；IRd上的傅立叶变换；有限傅里叶分析；Dirichlet定理。

【目录】

foreword
preface
chapter1.thegenesisoffourieranalysis
1thevibratingstring
1.1derivationofthewaveequation
1.2solutiontothewaveequation
1.3example：thepluckedstring
2theheatequation
2.1derivationoftheheatequation
2.2steady-stateheatequationinthedisc
3exercises
4problem

chapter2.basicpropertiesoffourierseries
1examplesandformulationoftheproblem
1.1maindefinitionsandsomeexamples
2uniquenessoffourierseries
3convolutions
4goodkernels
5cesaroandabelsummability：applicationstofourierseries
.5.1cesaromeansandsnmmation
5.2fejer'stheorem
5.3abelmeansands-ruination
5.4thepoissonkernelanddirichlet'sproblemintheunitdisc
6exercises
7problems

chapter3.convergenceoffourierseries
1mean-squareconvergenceoffourierseries
1.1vectorspacesandinnerproducts
1.2proofofmean-squareconvergence
2returntopointwiseconvergence
2.1alocalresult
2.2acontinuousfunctionwithdivergingfourierseries
3exercises
4problems

chapter4.someapplicationsoffourierseries
1theisoperimetricinequality
2weyl'sequidistributiontheorem
3acontinuousbutnowheredifferentiablefunction
4theheatequationonthecircle
5exercises
6problems

chapter5.thefouriertransformonr
1elementarytheoryofthefouriertransform
1.1integrationoffunctionsontherealline
1.2definitionofthefouriertransform
1.3theschwartzspace
1.4thefouriertransformon3
1.5thefourierinversion
1.6theplancherelformula
1.7extensiontofunctionsofmoderatedecrease
1.8theweierstrassapproximationtheorem
2applicationstosomepartialdifferentialequations
2.1thetime-dependentheatequationontherealline
2.2thesteady-stateheatequationintheupperhalf-plane
3thepoissonsummationformula
3.1thetaandzetafunctions
3.2heatkernels
3.3poissonkernels
4theheisenberguncertaintyprinciple
5exercises
6problems

chapter6.thefouriertransformonra
1preliminaries
1.1symmetries
1.2integrationonra
2elementarytheoryofthefouriertransform
3thewaveequationinrd×r
3.1solutionintermsoffouriertransforms
3.2thewaveequationinr3×r
3.3thewaveequationinr2×r：descent
4radialsymmetryandbesselfunctions
5theradontransformandsomeofitsapplications
5.1thex-raytransforminr2
5.2theradontransforminr3
5.3anoteaboutplanewaves
6exercises
7problems

chapter7.finitefourieranalysis
1fourieranalysisonz（n）
1.1thegroupz（n）
1.2fourierinversiontheoremandplancherelidentityonz（n）
1.3thefastfouriertransform
2fourieranalysisonfiniteabeliangroups
2.1abeliangroups
2.2characters
2.3theorthogonalityrelations
2.4charactersasatotalfamily
2.5fourierinversionandplancherelformula
3exercises
4problems

chapter8.dirichlet'stheorem
1alittleelementarynumbertheory
1.1thefundamentaltheoremofarithmetic
1.2theinfinitudeofprimes
2dirichlet'stheorem
2.1fourieranalysis，dirichletcharacters，andreduc-tionofthetheorem
2.2dirichletl-functions
3proofofthetheorem
3.1logarithms
3.2l-functions
3.3non-vanishingofthel-function
4exercises
5problems
appendix：integration
1definitionoftheriemannintegral
1.1basicproperties
1.2setsofmeasurezeroanddiscontinuitiesofinte-grablefunctions
2multipleintegrals
2.1theriemannintegralinrd
2.2repeatedintegrals
2.3thechangeofvariablesformula
2.4sphericalcoordinates
3improperintegrals.integrationoverrd
3.1integrationoffunctionsofmoderatedecrease
3.2repeatedintegrals
3.3sphericalcoordinates
notesandreferences
bibliography
symbolglossary