FOURIER ANALYSIS Fourieranalysisisasubjectthatwasborninphysicsbutgrewupinmathematics.Nowitispart ofthestandardrepertoireformathematicians,physicistsandengineers.Thisdiversityofinterest isoftenoverlooked,butinthismuch-lovedbook,TomKörnerprovidesashopwindowforsome oftheideas,techniquesandelegantresultsofFourieranalysis,andfortheirapplications.These range from number theory, numerical analysis, control theory and statistics, to earth science, astronomy and electrical engineering. The prerequisites are few (a reader with knowledge of second-orthird-yearundergraduatemathematicsshouldhavenodifficultyfollowingthetext), andthestyleislivelyandentertaining. This edition of Körner’s 1989 text includes a foreword written by Professor Terence Tao introducingittoanewgenerationoffans. T. W. Körner is Emeritus Professor of Fourier Analysis at the University of Cambridge. His otherbooksincludeThePleasuresofCounting(Cambridge,1996)andWhereDoNumbersCome From?(Cambridge,2019). CAMBRIDGE MATHEMATICALLIBRARY CambridgeUniversityPresshasalongandhonourablehistoryofpublishinginmathematics andcountsmanyclassicsofthemathematicalliteraturewithinitslist.Someofthesetitles havebeenoutofprintformanyyearsnowandyetthemethodswhichtheyespousearestill ofconsiderablerelevancetoday. TheCambridgeMathematicalLibraryprovidesaninexpensiveeditionofthesetitlesin adurablepaperbackformatandatapricethatwillmakethebooksattractivetoindividuals wishing to add them to their own personal libraries. Certain volumes in the series have a foreword,writtenbyaleadingexpertinthesubject,whichplacesthetitleinitshistorical andmathematicalcontext. A complete list of books in the series can be found at www.cambridge.org/mathematics. Recenttitlesincludethefollowing: AttractorsforSemigroupsandEvolutionEquations OLGAA.LADYZHENSKAYA FourierAnalysis T.W.KÖRNER TranscendentalNumberTheory ALANBAKER AnIntroductiontoSymbolicDynamicsandCoding(SecondEdition) DOUGLASLIND&BRIANMARCUS ReversibilityandStochasticNetworks F.P.KELLY TheGeometryofModuliSpacesofSheaves(SecondEdition) DANIELHUYBRECHTS&MANFREDLEHN SmoothCompactificationsofLocallySymmetricVarieties(SecondEdition) AVNERASH,DAVIDMUMFORD,MICHAELRAPOPORT& YUNG-SHENGTAI MarkovChainsandStochasticStability(SecondEdition) SEANMEYN&RICHARDL.TWEEDIE FOURIER ANALYSIS T. W. KÖRNER UniversityofCambridge With a Foreword by TERENCE TAO UniversityofCalifornia,LosAngeles UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre,NewDelhi–110025,India 103PenangRoad,#05–06/07,VisioncrestCommercial,Singapore238467 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781009230056 DOI:10.1017/9781009230063 ©CambridgeUniversityPress1988 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished1988 Firstpaperbackedition(withcorrections)1989 ReprintedwithForeword2022 AcataloguerecordforthispublicationisavailablefromtheBritishLibrary. ISBN978-1-009-23005-6Paperback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy ofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. ...mathematicalideasoriginateinempirics,althoughthegenealogyissometimes long and obscure. But, once they are so conceived, the subject begins to live a peculiar life of its own and is better compared to a creative one, governed by almost entirely aesthetical motivations, than to anything else and, in particular, to an empirical science. There is, however, a further point which, I believe, needs stressing. As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired by ideas coming from ‘reality’, it is beset with very grave dangers. It becomes more and more purely aestheticising, more and more purely 1’art pour 1’art. This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline is under the influence of men with an exceptionallywell-developedtaste.Butthereisagravedangerthatthesubjectwill develop along the line of least resistance, that the stream, so far from its source, will separate into a multitude of insignificant branches, and that the discipline will become a disorganised mass of details and complexities. In other words, at a great distance from its empirical source, or after much ‘abstract’ inbreeding, a mathematicalsubjectisindangerofdegeneration. vonNeumann(fromthefirstpaperinhiscollectedworks) Somecalculustricksarequiteeasy.Someareenormouslydifficult.Thefoolswho write the text books of advanced mathematics – and they are mostly clever fools – seldom take the trouble to show you how easy the easy calculations are. On the contrary, they seem to desire to impress you with their tremendous cleverness by goingaboutitinthemostdifficultway. Beingmyselfaremarkablystupidfellow,Ihavehadtounteachmyselfthedifficul- ties, and now beg to present to my fellow fools the parts that are not hard. Master thesethoroughly,andtherestwillfollow.Whatonefoolcando,anothercan. (fromCalculusMadeEasybySylvanusP.Thompson) ‘Now,’Herbiesays,‘waitaminute.Astorygoeswithit.’ (fromAStoryGoesWithItbyDamonRunyon) Contents ForewordbyTerenceTao pagexi Preface xiii PartI FourierSeries 1 1 Introduction 3 2 ProofofFeje´r’stheorem 6 3 Weyl’sequidistributiontheorem 11 4 TheWeierstrasspolynomialapproximationtheorem 15 5 AsecondproofofWeierstrass’stheorem 19 6 Hausdorff’smomentproblem 21 7 Theimportanceoflinearity 24 8 Compassandtides 28 9 Thesimplestconvergencetheorem 32 10 Therateofconvergence 35 11 Anowheredifferentiablefunction 38 12 Reactions 42 13 MonteCarlomethods 46 14 MathematicalBrownianmotion 50 15 Pointwiseconvergence 56 16 BehaviouratpointsofdiscontinuityI 59 17 BehaviouratpointsofdiscontinuityII 62 18 AFourierseriesdivergentatapoint 67 19 Pointwiseconvergence,theanswer 74 PartII SomeDifferentialEquations 77 20 Theundisturbeddampedoscillatordoesnotexplode 79 vii viii Contents 21 Thedisturbeddampedlinearoscillatordoesnotexplode 83 22 Transients 88 23 Thelineardampedoscillatorwithperiodicinput 93 24 Anon-linearoscillatorI 99 25 Anon-linearoscillatorII 104 26 Anon-linearoscillatorIII 113 27 Poissonsummation 116 28 Dirichlet’sproblemforthedisc 121 29 Potentialtheorywithsmoothnessassumptions 124 30 AnexampleofHadamard 131 31 Potentialtheorywithoutsmoothnessassumptions 134 PartIII OrthogonalSeries 143 32 MeansquareapproximationI 145 33 MeansquareapproximationII 150 34 Meansquareconvergence 155 35 TheisoperimetricproblemI 159 36 TheisoperimetricproblemII 166 37 TheSturm–LiouvilleequationI 170 38 Liouville 175 39 TheSturm–LiouvilleequationII 179 40 Orthogonalpolynomials 185 41 Gaussianquadrature 191 42 Linkages 197 43 TchebychevanduniformapproximationI 201 44 Theexistenceofthebestapproximation 207 45 TchebychevanduniformapproximationII 212 PartIV FourierTransforms 219 46 Introduction 221 47 ChangeintheorderofintegrationI 226 48 ChangeintheorderofintegrationII 230 49 Feje´r’stheoremforFouriertransforms 240 50 Sumsofindependentrandomvariables 245 51 Convolution 253 52 ConvolutiononT 259 53 Differentiationundertheintegral 265 54 LordKelvin 270 Contents ix 55 Theheatequation 274 56 TheageoftheearthI 282 57 TheageoftheearthII 285 58 TheageoftheearthIII 289 59 Weierstrass’sproofofWeierstrass’stheorem 292 60 Theinversionformula 295 61 Simplediscontinuities 300 62 Heatflowinasemi-infiniterod 308 63 Asecondapproach 315 64 Thewaveequation 324 65 ThetransatlanticcableI 332 66 ThetransatlanticcableII 335 67 UniquenessfortheheatequationI 338 68 UniquenessfortheheatequationII 344 69 Thelawoferrors 347 70 ThecentrallimittheoremI 349 71 ThecentrallimittheoremII 357 PartV FurtherDevelopments 363 72 Stabilityandcontrol 365 73 Instability 368 74 TheLaplacetransform 372 75 Deeperproperties 379 76 Polesandstability 386 77 Asimpletimedelayequation 395 78 Anexceptiontoarule 403 79 Manydimensions 407 80 Sumsofrandomvectors 413 81 Achisquaredtest 418 82 Haldaneonfraud 425 83 AnexampleofoutstandingstatisticaltreatmentI 429 84 AnexampleofoutstandingstatisticaltreatmentII 434 85 AnexampleofoutstandingstatisticaltreatmentIII 436 86 Willarandomwalkreturn? 443 87 WillaBrownianmotionreturn? 451 88 AnalyticmapsofBrownianmotion 455 89 WillaBrownianmotiontangle? 461 90 LaFamillePicardvaa´ MonteCarlo 467