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thetheoremsonextendingmappings–mainlyrestrictingourselvesto
Lipschitzmappingsforthesakeofsimplicity.Wealsoincludeavery
detailedproofofoneofthemoststrikingresultsonthesespaces,the
Baillonintersectiontheorem.Inthefinalsection,werevisitfixed-point
theoremsfromtheprecedingchapter,thistimeshowingtheirhyper-
convexanalogues.(Interestingly,whilethestatementsareoftenvery
similar,theproofsinthehyperconvexcasearecompletelydifferent.)
Thefourthchapteristhelongestoneinthebook,whichismainly
becauseitcontainswhatcouldalmostamounttotwochapters.We
startwiththebasicsofthetheoryofperiodicfunctions.Thisseems
somethingwell-knownfromanyvanillacalculuscourse,butthisim-
pressionisfarfromthetruth.Wepresentmanylittlegemsconcerning
periodicfunctions–lessthanhalfofthemisknownhalfaswellas
theydeserve,sotospeak.Wearemainlyinterestedintwoquestions:
howstrangeaperiodicfunctioncouldbe(leadingustothenotion
andsomepropertiesofso-calledmicroperiodicfunctions),andwhat
wecandotoperiodicfunctionssothattheirperiodicnatureisnot
lost(leadingustoconditionsunderwhiche.g.sums,products,deriva-
tivesorprimitivesofperiodicfunctionsarealsoperiodic).Afterthat
introducingsectionwefinallystarttalkingabouttherealtopic,which
isalmostperiodicfunctions.(Onemotivationtoconsiderthemisthe
factthatthesumoftwoperiodicfunctionsneednotbeperiodic,but
itmustbealmostperiodic.Wementionedawhileagothatperiodic
functionsdonotformalinearspace–buteveninnonlinearanalysis,
itisusefultohavesuchastructure,soitisnaturaltoconsiderthe
minimallinearspacecontainingallperiodicfunctions–thespaceof
almostperiodicones.)Wecontrastthemwiththeirperiodiccousins,
thendefineanotionofthemeanvaluewhichweusetobrieflysketch
thetheoryofFourierseriesofalmostperiodicfunctions.Weconclude
thechapterwithasectiononthespaceofallalmostperiodicfunctions
ontherealline,andanotheroneonalmostperiodicsolutionstosome
ordinarydifferentialequations.
Thefifthchaptercanbeviewedasanappendixtothefourthone,
asitdealswithtwoclassesofgeneralizationsofalmostperiodicfunc-
tions–theso-calledasymptoticallyandpseudoalmostperiodicfunctions.
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