Fourier Series, Fourier Transforms, and Function Spaces: A Second Course in Analysis (AMS/MAA Textbooks)

371 Pages·2020·1.904 MB·English

by  Tim Hsu

AMS / MAA TEXTBOOKS VOL 59
Fourier Series, Fourier Transforms,
and Function Spaces
A Second Course in Analysis
Tim Hsu

Fourier Series,
Fourier Transforms,
and Function Spaces:
A Second Course
in Analysis

AMS/MAA TEXTBOOKS
VOL 59
Fourier Series,
Fourier Transforms,
and Function Spaces:
A Second Course
in Analysis
Tim Hsu
MAATextbooksEditorialBoard
StanleyE.Seltzer,Editor
MatthiasBeck SuzanneLynneLarson JeffreyL.Stuart
DebraSusanCarney MichaelJ.McAsey RonD.Taylor,Jr.
HeatherAnnDye VirginiaA.Noonburg ElizabethThoren
WilliamRobertGreen ThomasC.Ratliff RuthVanderpool
2010MathematicsSubjectClassification.Primary26-01,42-01.
Foradditionalinformationandupdatesonthisbook,visit
www.ams.org/bookpages/text-59
LibraryofCongressCataloging-in-PublicationData
Names:Hsu,Tim(TimothyMing-Jeng),1969-author.
Title:Fourierseries,Fouriertransforms,andfunctionspaces:asecondcourseinanalysis/TimHsu.
Description: Providence,RhodeIsland: MAAPress,animprintoftheAmericanMathematicalSociety,
[2020]|Series:AMS/MAAtextbooks;volume59|Includesbibliographicalreferencesandindex.
Identifiers:LCCN2019040897|ISBN9781470451455(hardback)|ISBN9781470455194(ebook)
Subjects: LCSH:Fourieranalysis. |Fourierseries. |Fouriertransformations. |Functionspaces. |AMS:
Realfunctions[Seealso54C30]–Instructionalexposition(textbooks,tutorialpapers,etc.).|Harmonic
analysisonEuclideanspaces–Instructionalexposition(textbooks,tutorialpapers,etc.).
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Contents
Introduction xi
1 Overture 1
1.1 Mathematicalmotivation: Seriesoffunctions 1
1.2 Physicalmotivation: Acoustics 3
Part1 Complexfunctionsofarealvariable 7
2 Realandcomplexnumbers 9
2.1 Axiomsfortherealnumbers 9
2.2 Complexnumbers 13
2.3 Metricsandmetricspaces 14
2.4 Sequencesin𝐂andothermetricspaces 17
2.5 Completenessinmetricspaces 23
2.6 Thetopologyofmetricspaces 25
3 Complex-valuedcalculus 31
3.1 Continuityandlimits 32
3.2 Differentiation 40
3.3 TheRiemannintegral: Definition 45
3.4 TheRiemannintegral: Properties 52
3.5 TheFundamentalTheoremofCalculus 58
3.6 Otherresultsfromcalculus 62
4 Seriesoffunctions 73
4.1 Infiniteseries 74
4.2 Sequencesandseriesoffunctions 80
4.3 Uniformconvergence 84
4.4 Powerseries 95
4.5 Exponentialandtrigonometricfunctions 96
4.6 Moreaboutexponentialfunctions 101
4.7 TheSchwartzspace 104
4.8 Integrationon𝐑 105
vii
viii Contents
Part2 FourierseriesandHilbertspaces 113
5 Theideaofafunctionspace 115
5.1 Whichclockkeepsbettertime? 115
5.2 Functionspacesandmetrics 117
5.3 Dotproducts 121
6 Fourierseries 125
6.1 Fourierpolynomials 125
6.2 Fourierseries 127
6.3 RealFourierseries 132
6.4 ConvergenceofFourierseriesofdifferentiablefunctions 136
7 Hilbertspaces 139
7.1 Innerproductspaces 139
7.2 Normedspaces 144
7.3 Orthogonalsetsandbases 150
7.4 TheLebesgueintegral: Measurezero 156
7.5 TheLebesgueintegral: Axioms 162
7.6 Hilbertspaces 171
8 ConvergenceofFourierseries 177
8.1 Fourierseriesin𝐿2(𝑆1) 177
8.2 Convolutions 179
8.3 Dirackernels 180
8.4 ProofoftheInversionTheorem 185
8.5 ApplicationsofFourierseries 189
Part3 Operatorsanddifferentialequations 201
9 PDEsanddiagonalization 203
9.1 SomePDEsfromclassicalphysics 203
9.2 Schrödinger’sequation 208
9.3 Diagonalization 210
10 OperatorsonHilbertspaces 213
10.1 OperatorsonHilbertspaces 213
10.2 Hermitianandpositiveoperators 218
10.3 Eigenvectorsandeigenvalues 222
10.4 Eigenbases 225
11 Eigenbasesanddifferentialequations 229
11.1 Theheatequationonthecircle 230
11.2 Theeigenbasismethod 235
11.3 Thewaveequationonthecircle 237
11.4 Boundaryvalueproblems 244
11.5 Legendrepolynomials 250