Classical Theory of Fourier Series : Demystified and Generalised VIVEK V. RANE. The Institute of Science, 15, Madam Cama Road, Mumbai
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1 Clssil Thoy o Foi Sis : Dmystii Glis VIVEK V RANE Th Istitt o Si 5 Mm Cm Ro Mmbi-4 3 -mil ss : v_v_@yhoooi Abstt : Fo Rim itgbl tio o itvl o poit thi w i Foi Sis t th poit o th itvl big ot how wh th tio lmt boms xpssibl s Foi sis I this poss w lso glis th Foi thoy bigig i sh opts s iit Foi sis ight/lt h Foi sis W lso sm p th sbsis ospoig to tms i ithmti pogssio o th bsi Foi sis Ky wos : Foi sis lol lt/ight h iit; smmbility Rim-Stiltjs itgl Cs` o o ()
2 Clssil Thoy o Foi sis : Dmystii Glis VIVEK V RANE DEPARTMENT OF MATHEMATICS THE INSTITUTE OF SCIENCE 5 MADAM CAMA ROAD MUMBAI-4 3 INDIA -mil ss : v_v_@yhoooi Giv iit itvl poit i it Rim itgbl tio o th itvl w omlly i tigoomti (xpotil) Foi sis o th tio t th poit o th itvl w big ot i ll its glitis how wh th tio lmt boms xpssibl s th Foi sis W lso giv immit poo o th Cso o () smmbility o th Foi sis W shll goo l o Rim-Stiltjs itgtio thoy o whih w th to th boo o Apostol [] All this will b o sig th two ts mly ) gliz El s smmtio oml ) th boly ovgt sis xpssio i i o []
3 : : This ppoh bls s i tho [ 4 ] to itiy sty th Foi sis o th ivtivs (with spt to ist vibl) o Hwitz zt tio ζ ( s α) s tio o so vibl α o th it itvl [] Th w l with iit Foi sis o ζ ( s α ) o tiol vl o α i th it itvl Th w lso l with Lh s zt tio o simil lis This ppoh ws ist sot to i tho[3] I th pst otxt w lso th to tho [ ] itvl I [ ] Lt b tio i Rim itgbl o o lgth lt x b poit i [ ] th Foi sis o t th poit x o th itvl [ ] wit W i ix ( x) ~ i i (Expotil Foi Sis) x x w wit ( x) ( os b si ) ~ i ( ) os b ( ) si Foi sis) (Tigoomti Nottio : Fo l vibl w wit [ ] o its itgl pt W wit
4 ψ ( ) [ ] Not tht ( ) ψ i i is ot itg Fo x with i x < w wit ( x ) lim ( x ) wh > i this limit xists Similly o x with < x w wit (x) lim ( x ) wh > i th limit xists W ll ( x ) : 3 : th ight h limit o t x Similly w i ( x ) th lt h limit o t th poit x I th s o ithmti ms o giv sis ovgs th th giv sis is si to b Cs` o smmbl o ( ) smmbl Usig o ppoh w gliz th opt o Foi sis to lol Foi sis lol ight h/lt h Foi sis iit Foi sis I t w pov th ollowig Thom : Lt b tio i Rim itgbl o th itvl [ ] o lgth lt ψ ( ) [ ] Lt b poit i ( ) I) I ( ) xists th w hv ( ) ψ
5 II) I ( ) xists th w hv ( ) ψ II) I both ( ) ( ) xists i ( ) th w hv ( ) ( ) ( ) ( ) ψ : 4 : I itio i w ithg i th Rim-Stiltjs itgl ψ ( ) tht is i i i i i th w hv th Foi sis i ( ) ( )
6 Othwis th ight h Foi sis ovgs to ( ) ( ) i () ss Not : I th tio mits o ithg o th sh tio will b ll goo tio o th itvl [ ] I is tio o bo vitio o [ ] th th Rim Stiltjs itgl bhvs li Rim itgl ths b ithg O th oth h i is itibl with its ivtiv ' Rim o Lbsg itgbl o I [ ] th th Rim-Stiltjs itgl ψ boms Rim itgl ( ) ' ( ) ithgbl ψ ths : 5 :
7 Coolly : I) Lt b Rim itgbl o itvl [ ] lt ( ) xist Th w hv ( ) ( ) ψ i is goo o th itvl ( ) th w hv lt h lol Foi i i sis mly ( ) ( ) II) Lt b Rim itgbl o th itvl [ ] lt ( ) xist Th ( ) ( ) ( ) ψ I itio i is goo o [ ] w hv ight h lol Foi sis i i mly ( ) ( ) III) Lt b b Rim itgbl o th itvl [ ] lt both ( ) ( ) xist Th w hv ψ ( ) ( ) ( ) I itio i is goo o [ ] i i ( ) ( ) ( ) w hv th lol Foi sis mly
8 Bo sttig o xt Thoms w ito itiol ottios Aitiol Nottios : Fo giv itg giv itg with ψ osi th tio ( ) i i ( mo ) W shll show i wht ollows tht ( ) ψ is stp tio o < I is itg th : 6 : ψ ( ) i i m i { ( m m ) } m m i i m m m m ( ) sy Nxt w stt o xt Thoms Thom : Lt b Rim itgbl o th it itvl [ ] with Fo itg lt b otios t th tiol poits o Lt b itg with
9 Th ( ) b i with ( ) ( ) b ψ o ( ) ( ) b ψ Rm : Not tht o th giv itg th oiits b s ipt o th itg wh vis ov th st { 3} Ths i x is tiol mb i th it itvl [ ] th th two Foi sis t ix x mly ) th sl iiit Foi sis x ~ with i ( ) th iit Foi sis molo mly b i Howv i b wh b itgs th i b b b wh ( b ) : 7 : is ipt o th itg b wh b vis ov { } Ths w hv iit Foi sis molo
10 W shll pov Thom sig o Thom 3 whih w stt blow ψ Thom 3 : Fo th tio ( ) i i ( mo ) is pioi stp tio with pio o < w hv ψ ( ) ( ) ' wh sh ov iits tht th tm ospoig to is to b hlv i is o th om R o som itg R Not : W hv ( o ) ( [ ] ) ψ o o-itgl Bo w giv th poos o o Thoms w stt w lmms Lmm : Lt α b tio i o itvl [ b] with jmp α t x wh x x x th simpl isotiitis o th tio α Lt b tio i o [ b] sh tht ot both α hv isotiitis om lt o om ight t h x Th b α xists w hv b α x α : 8 :
11 Rm : Lmm is th Thom 7 o th Chpt 7 o Apostol s boo [] H α ( x ) α ( x ) α i < x < b x b th α α ( x ) α ( x ) α I x α α ( x ) α ( x ) ; i Nxt w stt o Lmm Lmm : (Glis El s smmtio oml) : Lt b Rimitgbl o th itvl [ b] sh tht is otios om lt t vy itg with α < b Th ( ) ( [ ] ) ( ) ( [ ] ) ( b) ( b [ b] ) < b b b Rm : Th poo ollows om Lmm Lmm : W hv o > > ) I ( ) xists th ( ) ( ) [ ] o < ) I ( ) xists th ( ) ( ) [ ] o < I otios t w hv ( ) [ ] o < < Not : Th lmm ollows om Lmm
12 : 9 : Lmm 3 (Itgtio by pts) : Lt th Rim-Stiltjs itgl α xist o th itvl [ b] Th α b lso xists w hv b α ( b) α ( b) α b α Poo o Thom : Lt b th poit i th op itvl ( ) lt ( ) xist Th by Lmm w hv ( ) ( ) [ ] ( ) ( ) ([ ]) [ ( )( [ ])] ( [ ]) ( ) o itgtio by pts
13 Ths [ ] Witig ψ [ ] w hv ψ Ths ψ Similly i xists w hv [ ] [ ] : : [ ] [ ] [ ] o itgtio by pts Ths [ ] [ ] [ ]
14 This givs ψ Ths ψ Ths ψ I th This givs ψ i Nxt si ψ si i ithgbl This givs ψ os si [ ] [ ] os ψ ψ
15 : : os os os os ) ( i i i Ths i i povi w ithg i th Rim-Stiltjs itgl si Nxt w pov tht th s o ithmti ms o th Foi sis o t th poit ovgs to i both xist Lt S i i
16 si os i [ ] si si { } si si si s : : Ths { } S si si os Not tht This givs S si si ψ sy wh ψ si Lt N N N S S N ψ σ
17 N ( ) ( ) ( ) ψ ( ) N ψ Ths N ( N ) N ( ) lim N lim σ N N ψ povi th limit xists N N N Si lim ( ) ψ ( ) ψ w hv σ ( N ) lim N ( ) N ψ Now s ( ) w hv ψ ( ) ( ) ( ) Ths w hv N ( N ) ( ) ( ) ( ) σ lim N Ths th Foi sis o t th poit i [ ] ovgs to ( ) ( ) i () ss : 3 :
18 Poo o Thom : Lt x b tiol mb sh tht < wh itgs As w hv i ( ) ( ) ( ) ( ) i ψ i i ( ) i ( ) ( mo ) i i ( mo ) i i ( ) ψ ( ) ( ) ( ) ψ ( ) povi ( ) ψ xists o h Howv th sttmt o Thom 3 shows tht ( ) ψ is stp tio o h with simpl isotiitis om lt t th poits o wh is otios om lt h ( ) ψ xists o h
19 Ths o i i ψ b b ψ sy wh ( ) ( o ) b ψ Witig b b o w hv ( ) b b b i Poo o Thom 3 : Lt b itgs with < Lt ( ) { } ( i i m m i ) m m m m Nxt lt < : 4 : < x Fo lg positiv itg M lt S ( x) M M i ( m ) x m M i ( m ) b th sm o th ist ( M ) tms Not x i ( m ) x i ( m ) i ( m ) i m S Ths M i ( m ) x M mm x M mm i ( m ) x x xi si ( M ) ( i ) i ( M ) i ( M ) ( si )
20 Lt > b siitly smll Lt R b th lgst itg sh tht R x I R < x th lt I R R th lt I ( x x ) I x R so tht I is isjoit io o itvls Lt I I I Ths I is lso isjoit io o itvls Ths x si ( M ) i si ( M ) si ( si ) I I i Not tht o I th omito si is bo wy om zo Cosi L wh L is o o th o-ovlppig sb-itvls whos io is I Lt < b th poits o L i Th L M si si os ( M ) ( M ) i si [ ] I viw o th to ( ) itgl h s M : 5 : i os ( M ) M i ( os si ) M ) M si M i omitos both th itgt pt th Nxt w vlt i si ( M ) ( si ) s M I
21 i si ( M ) H osi ( ) si Witig v w hv th bov itgl i ( v ) ( M ) ( v ) si v si v ( ) i v M v si v si v i iv si si ( M )( v ) v v i ( M ) i si si i i si ( M ) ( ) i i os si si si i ( M ) os si ( si ) ( M ) Usig th slt mly i tio g is o bo vitio o itvl [ ] th lim siαt g g t t t ( ) α i th light o th t tht lim os si w hv i si ( M ) lim si M i Similly w show i si ( M ) lim si M : 6 :
22 x i i si ( M ) lim M x si i x is tio o th om Lttig M w gt ψ ( x ) lim S M ( x) i ( m ) φ ( x ) M m wh o < i ' φ x H sh i ' < x w hv x iits th lst tm is to b hlv i x o som itg This omplts th Poo o Thom 3 Rs [] TM Apostol Mthmtil Alysis So Eitio (974) Aiso- Wsly [] VV R Alogs o El Poisso smmtio oml Po Ii A Si Mth Si 3 o3 3- (3) [3] VV R A Uii Appoh to lysis Nmb thoy Ht 8 (l985) Mthmti Gottigsis (pp 48-57) [4] VV R Foi Sis o Divtivs o Hwitz Lh s Zt Ftio ( pptio)
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