These symmetric functions are usually quite explicit such as a trigonometric function sin nx or cos nx and are often associated with physical concepts such as frequency or energy What symmetric means here will be left vague but it will usually be ID: 23814
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2TERENCETAOnotethatthepreviousdecompositionintoevenandoddfunctionswassimplythen=2specialcaseofthisformula.ThegroupassociatedtothisFouriertransformisthenthrootsofunityfe2ik=n:0kn�1g,witheachrootofunitye2ik=nassociatedwiththerotationz7!e2ik=nzonthecomplexplane.Movingnowtothecaseofinnitegroups,considerafunctionf:T!CdenedontheunitcircleT:=fz2C:jzj=1g;toavoidtechnicalissuesletusassumethatfissmooth(i.e.innitelydi erentiable).Observethatiffisamonomialfunctionf(z)=cnznforsomeintegern,thenfwillobeytherotationalsymmetryofordernf(eiz)=einf(z)forallcomplexnumberszandallphases.Itshouldnowbenosurprisethatanarbitrarysmoothfunctionfcanbeexpressedasasuperpositionofsuchrotationallysymmetricfunctions:f(z)=1Xn=�1^f(n)zn;where^f(n):=1 2Z20f(ei)e�ind:Thisformulacanbethoughtofasthelimitingcasen!1ofthepreviousdecom-position,restrictedtotheunitcircle.ItalsogeneralizestheTaylorseriesexpansionf(z)=1Xn=0anzn;wherean=1 2iZjzj=1f(z) zn+1dzfromcomplexanalysis,whenfisacomplexanalyticfunctionontheclosedunitdiskfz2C:jzj1g;indeedthereareverystronglinksbetweenFourieranalysisandcomplexanalysis.Thecomplexnumbers^f(n)areknownastheFouriercoecientsoffatgivenfrequenciesormodesn.Whenfissmooth,thenthesecoecientsdecayveryquicklyandthereisnoprobleminestablishingconvergenceoftheFourierseriesP1n=�1^f(n)zn.Theissuebecomesmoresubtleiffisnotsmooth(forinstance,ifitismerelycontinuous),andonehastospecifythenatureofthisconvergence;infactasignicantportionofharmonicanalysisisdevotedtothesetypesofquestions,andindevelopingtools(andestimates)thatcanaddressthem.ThegroupassociatedwiththisFourieranalysisisthecirclegroupT.Butthereisnowalsoasecondgroupwhichisimportanthere,whichistheintegergroupZ;thisgroupindexesthetypeofsymmetriesavailableontheoriginalgroupT(foreachintegern2Z,onehasanotionofarotationallysymmetricfunctionofordernonT),andisknownasthePontryagindualtoT.(Inthepreviousexamples,theunderlyinggroupanditsPontryagindualwerethesame.)Inthetheoryofpartialdi erentialequationsandinrelatedareasofharmonicanalysis,themostimportantFouriertransformisthatonaEuclideanspaceRd.Amongallfunctionsf:Rd!C,therearetheplanewavesf(x)=ce2ix,where2Rdisavector(knownasthefrequencyoftheplanewave),xisthedotproductbetweenthepositionxandthefrequency,andcisacomplexnumber(whosemagnitudeistheamplitudeoftheplanewave).Itturnsoutthatifafunctionfissuciently\nice"(e.g.smoothandrapidlydecreasing),thenitcanberepresenteduniquelyasthesuperpositionofplanewaves,wherea\superposition"isinterpreted 4TERENCETAOHerewehaveinterchangedtheLaplacianwithanintegral;thiscanberigourouslyjustiedforsuitablynicef,butweomitthedetails.Sincefhasauniquerepre-sentationRRdcf()e2ixd,weconcludethatcf()=(�4jj2)^f();thisidentitycanalsobederiveddirectlyfromthedenitionoftheFouriertransformandfromintegrationbyparts.ThisidentityshowsthattheFouriertransformdiagonalizestheLaplacian;theoperationoftakingtheLaplacian,whenviewedusingtheFouriertransform,isnothingmorethanamultiplicationoperatorbyanexplicitmultiplier,inthiscasethefunction�4jj2;thisquantitycanalsobeinterpretedastheenergylevelassociated4tothefrequency.Inotherwords,theLaplaciancanbeviewedasaFouriermultiplier.ThisviewpointallowsonetomanipulatetheLaplacianveryeasily.Forinstance,wecaniteratetheaboveformulatocomputehigherpowersoftheLaplacian:dnf()=(�4jj2)n^f()forn=0;1;2;:::Indeed,thisnowsuggestsawaytodevelopmoregeneralfunctionsoftheLaplacian,forinstanceasquareroot:\p �f()=p 4jj2^f():Thisleadstothetheoryoffractionaldi erentialoperators(whichareinturnaspecialcaseofpseudodi erentialoperators),aswellasthemoregeneraltheoryoffunctionalcalculus,inwhichonestartswithagivenoperator(suchastheLapla-cian)andthenstudiesvariousfunctionsofthatoperator,suchassquareroots,exponentials,inverses,andsoforth.Astheabovediscussionshows,theFouriertransformcanbeusedtodevelopanumberofinterestingoperations,whichhaveparticularimportanceinthetheoryofdi erentialequations.Toanalyzetheseoperationse ectivelyoneneedsvariousestimatesontheFouriertransform,forinstanceknowinghowthesizeofafunctionf(insomenorm)relatestothesizeofitsFouriertransform(perhapsinadi er-entnorm).OneparticularlyimportantandstrikingestimateofthistypeisthePlancherelidentityZRdjf(x)j2dx=ZRdj^f()j2dwhichshows,amongotherthings,thattheFouriertransformisinfactaunitaryoperation,andsoonecanviewthefrequencyspacerepresentationofafunctionasbeinginsomesensea\rotation"ofthephysicalspacerepresentation.DevelopingfurtherestimatesrelatedtotheFouriertransformandassociatedoperatorsisamajorcomponentofharmonicanalysis.AvariantofthePlancherelidentityistheconvolutionformulaZRdf(y)g(x�y)dy=ZRd^f()^g()e2ixd:Thisformulaallowsonetoanalyzetheconvolutionfg(x):=RRdf(y)g(x�y)dyoftwofunctionsf;gintermsoftheirFouriertransform;inparticular,ifforghavesmallFouriercoecientsthenweexpecttheirconvolutionfgtoalsobesmall. 4Whentakingthisperspective,itiscustomarytoreplaceby�inordertomaketheenergiespositive.