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AcademicPress
FourierSeries
JamesS.Walker
DepartmentofMathematics
UniversityofWisconsin-EauClaire
EauClaire,WI54702-4004
Phone:715-836-3301
Fax:715-836-2924
e-mail:walkerjs@uwec.edu 1
I.Introduction
II.Historicalbackground
III.DefinitionofFourierseries
IV.ConvergenceofFourierseries
V.Convergenceinnorm
VI.SummabilityofFourierseries
VII.GeneralizedFourierseries
VIII.DiscreteFourierseries
IX.Conclusion
GLOSSARY
Boundedvariation:Afunction
,theinequality? ???????is ?????).Examples:The
Continuousfunction:If
?,thenthefunction ?iscontinuous atthepoint ?.Suchapointiscalledacontinuitypointfor ?.Afunctionwhichis
Lebesguemeasurezero:Aset
zeroif,foreach ?????,thereexistsacollection? ?ofopenintervalssuch that ???and? ??????.Examples:Allfinitesets,andall
Oddandevenfunctions:Afunction
?isoddif ?forall ?inits domain.Afunction ?isevenif ?forall ?initsdomain.
One-sidedlimits:
???and ?denotelimitsof ??????as ?tendsto ?fromthe leftandright,respectively.
Periodicfunction:Afunction
?holdsforall ?.Example: ?isperiodicwithperiod
I.Introduction
Fourierseries3
andlocaltrigonometricanalysis.
II.Historicalbackground
?century. ?ofathinwireoflength ?,stretchedbetween ???and ?,witha constantzerotemperatureattheends: ???and ???.Heproposed thattheinitialtemperature ?couldbeexpandedinaseriesofsine functions: ?(1) with ????(2) trary"function ?satisfies ??(3) forsolvingboundaryvalueproblems. sourceofknowledge."
III.DefinitionofFourierseries
animalheartbeats.
Afunction
?issaidtohaveperiod?if ?forall ?.For ofscale
Ifthefunction
?hasperiod? ?,thenitsFourierseriesis ?(4) withFouriercoefficients ?,and? ?definedbytheintegrals ??(5) ????(6) ????(7) If ?isinitiallydefinedovertheinterval??? ??,thenitcanbeextendedto? (asanoddfunction)byletting ?,andthenextendedperiodically withperiod thesineseriesinEq.s(1)and(2),because ?????,each? ????,andeach? ?in
Eq.(7)isequaltothe
?inEq.(2).]
Fourierseries5
thecomplexexponential ??satisfies ?.Hence ??????(8) with ??definedforallintegers?by ???(9)
Theseriesin(8)isusuallywrittenintheform
(10)
Wenowconsideracoupleofexamples.First,let
??bedefinedover? ??by ?if if andhaveperiod? ?.Thegraphof ??isshowninFig.1;itiscalledasquarewave inelectriccircuittheory.Theconstant ??is
While,for??????,
Figure1:Squarewave.
Thus,theFourierseriesforthissquarewaveis
??(11)
Second,let
?over? ??andhaveperiod? ?.SeeFig.2.Weshall ????and for ?????,is
Fourierseries7
Figure2:Parabolicwave.
??(12) ??and ??respec- tively,inSectionIV. hasasmallestperiodof? ??????isrepeated???? coefficient ??canbeinterpretedasacorrelationbetween ?andacomplexexpo- nentialwithapreciselylocatedfrequencyof ?.Thusthewholecollectionof theseintegrals,forallintegers ?,specifiesthefrequencycontentof ?overtheset offrequencies ???.Iftheseriesin(10)convergesto ?,i.e.,ifwecan write ???(13) then ??having frequency ?andamplitude if if ???.(14) derivationofEq.(9).MultiplyingEq.(13)by ?????andintegratingterm-by-term from ?to ?,weobtain
Bytheorthogonalityproperty,thisleadsto
??inEq.(9).
Theorem1(Bessel'sInequality)If
??isfinite,then ???(15) obtain ?(16)
Fourierseries9
Thus,forall
??(17) ?hasfiniteenergy,inthe ???(18) holdswhenever
Theorem2(Riemann-LebesgueLemma)If
??isfinite,thenEq.(18) holds. section. andtheRiemann-Lebesguelemma,see[7]or[12]
IV.ConvergenceofFourierseries
andtakeupthethirdtypeinthenextsection. convergeto ?.Thatis,does????? ?holdinsomesense? holdsforeachfixed ?-value ???.If????? ?doesequal ?,thenwesay thattheFourierseriesfor ?convergesto ?at point ???if,forsomepositiveconstant?, ????(19) holdsforall ?near ???(i.e., ??forsome ???).Itiseasytosee,for instance,thatthesquarewavefunction ??isLipschitzatallofitscontinuitypoints. are ?when ???,thisinequalityisequivalentto ?(20) forall ?near ???(and quotientsof ?(i.e.,theslopesofitssecants)near ??arebounded.Withthisin- ??isLipschitzatallpoints.
Moregenerally,if
?hasaderivativeat ???(orevenjustleft-handandright-hand derivatives),then ?isLipschitzat
Theorem3Suppose
?hasperiod? ?,that ??isfinite,andthat ?is
Lipschitzat
???.ThentheFourierseriesfor ?convergesto ?at
Toprovethistheorem,weassumethat
????.Thereisnolossofgeneralityin ??????from ?.Definethe function ?by? ?.Thisfunction?hasperiod? ?.Further- more, ??isfinite,becausethequotient ?isboundedin magnitudefor ?near ???.Infact,forsuch
Fourierseries11
and ofthederivativeof ?at
Ifwelet
??denotethe?? ?Fouriercoefficientfor? ?,thenwehave because ?.Thepartialsum? ?then telescopes: Since ?as ???.Thiscompletestheproof.
Itshouldbenotedthatforthesquarewave
??andtheparabolicwave ??,it ??isfiniteforthe function ?as followsfromBessel'sinequalityfor?. convergesto seriesfortheparabolicwave ??convergesto ??atallpoints.Whilethismaysettle oftheirconvergencetothesewaves. ?????superimposed ?as rate.Thepartialsum ?????differssignificantlyfrom ??.Nearthesquarewave's overshootthesquarewave'svalueof ?,tendingtoalimitofabout? ???.Thepartial rangeoffrequencies-cannotuse ?????,oranypartialsum? ?,toproduceasquare
Figure3:Fourierseriespartialsum?
?????superimposedonsquarewave. sumlike wave ??tendtozeromore rapidly(ataratecomparableto ?).Becauseofthis,thepartialsum? ?????for sums
WesaythattheFourierseriesforafunction
?convergesuniformlyto ?if ?(21)
Thisequationsaysthat,forlargeenough
,wecanhavethemaximumdistance betweenthegraphsof ?and? ofthisfortheparabolicwave.
Fourierseries13
Consequently
andthusEq.(21)holdsfortheparabolicwave theorem.Weshallsaythat usingthesameconstant
Theorem4Supposethat
?hasperiod? ?andisuniformlyLipschitzatallpoints, thentheFourierseriesfor ?convergesuniformlyto
Theorem4appliestotheparabolicwave
??,butitdoesnotapplytothesquare wave ??.Infact,theFourierseriesfor ??cannotconvergeuniformlyto ??.That
Figure5:Fourierseriespartialsum?
?????forparabolicwave. ousfunctions(likethepartialsums ?)mustbeacontinuousfunction(which let.Thisintegralformis ?(22) withkernel ?definedby ?(23) [7],[12],or[16]).Thekernel ?iscalledDirichlet'skernel.InFig.6wehave graphed
Fourierseries15
Figure6:Dirichlet'skernel
FromEq.(23)wecanseethatthevalueof
?followsfromcancellationofsigned ?(seeFig.6)is significantlygreaterthan ?(about? ???invalue). ??,Eq.(22)becomes ??(24) As ?rangesfrom ?to ?,thisformulashowsthat? ?isproportionaltothe signedareaof ?overanintervaloflength ?centeredat ?.ByexaminingFig.6, whichisatypicalgraphfor partialsums alsoresultsfromEq.(24).When ?nearsajumpdiscontinuity,thecentrallobe of whichovershootsthevalueof ?for ??byabout???
Theorem5If
?hasperiod? ?andhasboundedvariationon????? ??,thenthe
Fourierseriesfor
directlybysubstitutionof ????intotheseriesin(11)]. ?function(for which everypoint.
V.Convergenceinnorm
gencein ?-norm(alsocalled? ?-convergence.Thereisalsoanin- terpretationofquotesdbs_dbs20.pdfusesText_26
Fourier Series, Fourier Transform and their Applications to