傅立叶分析导论

　　《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。

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