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12
I
Likeinthecaseofalmostperiodicfunctions,westartwiththedefini-
tion,andthenexaminesomebasicpropertiesofthesefunctionsfirst
andofthespaceofallofthemnext.Sincewedidnotwanttodevote
halfofthebooktoonesubject,thesetopicsareonlysketched,sothat
thischapteristheshortestone.Still,wecrammedashortsectionon
applications(alsotodifferentialequations)atitsend.
Thelastflregular”chapterisdevotedtotheso-calledfunctionsof
bounded(Jordan)variation.(Likeinthecaseofalmostperiodicfunctions,
theyariseverynaturallyastheminimallinearspaceincludingtheset
ofallmonotonefunctions.)Theysometimesmakeashortappearance
inthoseintroductorycalculustextbookswhichincludeasectionon
theRiemann–Stieltjesintegralandthenarepromptlyforgotten.This
iskindofunfair,sincetheyareaveryinterestingtopic.Whiletheir
definitionisfairlysimple,itleadstoawidearrayofproperties–some
surprising,someenlightening.Weroughlyrepeatthecourseknown
fromthechapteronalmostperiodicfunctions,althoughwithabitof
variation(seewhatwedidhere?)intheorder(seetheintroduction
ofthatchaptertolearnwhy).Wealsospendquiteafewpageson
showingthreevariouswaysfunctionsofboundedvariationcanbe
decomposedintosimplerflbuildingblocks”.Ofcourse,wehadto
includeasectionaboutapplications–toequations(integralonesthis
time)andclassicalFourierseries.
Bytheway,thislastchapterwasinitiallymeanttoonlyskimthe
theoryoffunctionsofboundedvariation,andbedevotedmainly
tooneofthemanygeneralizationsofthatnotion,thefunctionsof
bounded
Λ
-variation.(Thisclassoffunctionsiswiderthanthefunc-
tionsofboundedJordanvariation,andappearsnaturallyinthetheory
ofFourierseries.)Duringwriting,however,itturnedoutthatour
roadleadssomewhereelse:wefiguredthat
Λ
-variation,whilenot
necessarilytoohardtoincludeinourbook,isdefinitelytootechnical,
anddecidedthatwehadbettergiveallreadersasolidgraspofthe
classicalvariationfirst,andthenpointthemorecuriousonestoref-
erencesonotherconceptsofvariation.(Eventhoughwerestricted
ourselvestoaveryclassicalnotionofJordanvariation,wemanagedto
