Ideas of Lebesgue and Perron Integration in Uniqueness of Fourier and Trigonometric Series By Ng Tze Beng

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1 Ides of Lebesgue d Perro Itegrtio i Uiqueess of Fourier d Trigoometric Series By Ng Tze Beg Tis rticle is bout te ides ledig to te uiqueess of coverget trigoometric series We loo t te ides ivolved we te limit fuctio of trigoometric series is Lebesgue itegrble Troug te use of Perro s tecique, we crcterize Lebesgue itegrbility by Perro s mjor d mior fuctios Troug tis d de l Vllée- Poussi s mjort d miort fuctios d Riem s ide of pssig from te symmetric secod derivtive to te trigoometric series, ide ow clled R- summbility, we deduce te uiqueess of Fourier Lebesgue series s stted i Teorem We te trigoometric series eed ot coverge to Lebegsue itegrble fuctio but is everywere coverget, buildig o te ide of Perro s mjor d mior fuctios but usig te secod symmetric derivtive, Jmes s J-mjor d J-mior fuctios re used to itroduce te ide of P itegrl to prove te uiqueess of everywere coverget trigoometric series, were te coefficiets re ow recovered by te P itegrl Itroductio Cosider te trigoometric series b cos( ) si( ) (A) It is our im to uderstd te proof of te uiqueess of te Fourier series of te sum fuctio of (A), tt is, te proof of te followig fudmetl result bout Fourier series Teorem If te series (A) coverges ecept i eumerble set E to fiite d itegrble fuctio f, te it is te Fourier series of f Our pproc would be tt of Riem d uses Perro s mjor d mior fuctios togeter wit te geerlized secod derivtive We te series (A) coverges everywere to o Lebesgue itegrble fuctio f, te coefficiets re ot recoverble by Lebesgue itegrtio We preset solutio by R D Jmes usig is P itegrls We sll elborte te ides iside tree sectios Sectio A cosiders crcteriztio of te Lebesgue itegrl i terms of lower d upper semi-

2 cotiuous fuctio Properties of tese fuctios tt re useful will be elborted d te ides of upper d lower derivtes will be itroduced to provide mes of obtiig mjor d mior fuctios Sectio B te cosiders te ide of te geerlized symmetric secod derivtive of fuctio, its properties d Riem s ide i te proof Sectio C cotis te uiqueess teorems d teir proofs We itroduce ere te ide of R D Jmes P itegrl d use tis to prove te uiqueess of everywere covergece trigoometric series Sectio A Semi-cotiuous Fuctio, Mjor d Mior Fuctios d Lebesgue Teory Lower d upper semi-cotiuous fuctios Let R* be te eteded rel umbers Suppose f : [, b] R* is eteded rel vlued fuctio Let m ( ) limif f ( t) : t B(, ) [, b] f d M ( ) lim sup f ( t) : t B(, ) [, b], were B(, ) (, ) f Plily, by te defiitio of lim if d lim sup, for ll i [, b], m ( ) f ( ) M ( ) f f Defiitio Let c be i [, b] Te fuctio f is sid to be lower semicotiuous t c if m ( c) f ( c) f is sid to be upper semi-cotiuous t c if M ( c) f ( c) f f Terefore, fiite vlued fuctio is cotiuous t c if d oly if it is bot lower d upper semi-cotiuous t c Note tt just vig oe sided semicotiuity t c does ot imply cotiuity t c To wor wit lower d upper semi-cotiuous fuctio, it is useful to use equivlet form of tis property Te followig teorem gives equivlet defiitio of lower d upper semi-cotiuity

3 Teorem Suppose f : [, b] R* is eteded rel vlued fuctio Let c be i [, b] () f is lower semi-cotiuous t c if d oly if for ec < f (c), tere eists > suc tt f () > for ll B(c, ) [, b] (b) Suppose f (c) is fiite Te fuctio f is lower semi-cotiuous t c if d oly if for ec >, tere eists > suc tt f () > f (c) for ll B(c, ) [, b] (c) f is upper semi-cotiuous t c if d oly if for ec > f (c), tere eists > suc tt f () < for ll B(c, ) [, b] (b) Suppose f (c) is fiite Te fuctio f is upper semi-cotiuous t c if d oly if for ec >, tere eists > suc tt f () < f (c) + for ll B(c, ) [, b] Proof We prove oly prts () d (b) Prts (c) d (d) re similrly proved () If f (c) =, te we ve otig to prove s plily m ( c) f ( c) If m ( c) f ( c), te by defiitio of m () c, for ec < f (c) =, f tere eists > suc tt f () > for ll B(c, ) [, b] If f (c) is fiite, te by defiitio of m ( c) f ( c) for ec < f (c), tere eists > suc tt f () > for ll B(c, ) [, b] f (b) If f (c) is fiite, te te = f (c) < f (c) Prt (b) te follows from prt () f f Te et property cocers etreme vlues of semi-cotiuous fuctio o [, b] Teorem 3 Suppose f : [, b] R* is eteded rel vlued fuctio (i) If f is lower semi-cotiuous o [, b], te f ssumes its miimum vlue (ii) If f is upper semi-cotiuous o [, b], te f ssumes its mimum vlue 3

4 Proof (i) Suppose f is lower semi-cotiuous o [, b] Let m if f ( ) : [, b] If m = +, te f is costt fuctio tig + d we ve otig to prove If m =, te tere eists sequece ( ) i [, b], suc tt f ( ) If m is fiite, te tere eists sequece ( ) i [, b], suc tt f ( ) m I bot cses, by te Bolzo-Weierstrss Teorem, ( ) s covergece subsequece ( ) Suppose d Te if m is fiite, d so f (d) = m m if{ f ( ) : [, b]} f ( d) m ( d) limif f ( ) lim f ( ) m, If, m =, te lim f( ), d f ( d) m ( d) limif f ( t) : t B( d, ) [, b] m f Te proof of prt (ii) is similr d is omitted For fiite fuctio f we ve te followig obvious corollry Corollry 4 Suppose f : [, b] R is rel vlued fuctio (i) If f is lower semi-cotiuous o [, b], te f is bouded below (ii) If f is upper semi-cotiuous o [, b], te f is bouded bove f Remr A fiite-vlued semi-cotiuous fuctio is Bire clss oe fuctio, ie, it is te poitwise limit of sequece of cotiuous fuctio 4

5 Te et result reltes Lebesgue itegrble fuctio wit semi-cotiuous fuctios Teorem 5 Suppose f : [, b] R* is Lebesgue itegrble For ec >, tere eist fuctios u d v suc tt (i) u is lower semi-cotiuous o [, b] d v is upper semi-cotiuous o [, b] ; (ii) for ll i [, b], u() >, u() f () v() d v() < ; (iii) u d v re Lebesgue itegrble o [, b] d b b b u f v To estblis Teorem 5 we sll use te followig crcteriztio of semicotiuous fuctio Teorem 6 Suppose f : [, b] R* is eteded rel vlued fuctio () f is lower semi-cotiuous o [, b] if d oly if te set { [, b]: f ( ) } f ([, ]) is closed for every rel umber (b) f is upper semi-cotiuous o [, b] if d oly if te set Proof { [, b]: f ( ) } f ([, ]) is closed for every rel umber We prove prt () oly Te proof for prt (b) is similr Suppose f is lower semi-cotiuous o [, b] Let be y rel umber Let 5

6 B b f f { [, ]: ( ) } ([, ]) If B =, te B is closed Now ssume B Let c be limit poit of B Te c is i [, b] We sll sow tt c is i B, ie, f (c) If f (c ) =, te obviously, c is i B We ow ssume tt f (c ) > By Teorem prt(), for y < f (c), tere eists > suc tt f () > for ll B(c, ) [, b] Sice c is limit poit of B, tere eists poit d i B(c, ) B {c} suc tt < f (d) Tis sows tt is upper boud of te set (, f (c)) Terefore, f (c) Hece c is i B Tis sows tt B is closed Coversely, suppose B is closed for every rel umber Te c i [, b] If f (c) =, te m ( c) f ( c) d so f is lower semi-cotiuous t c f Assume ow f (c) > Let < f (c) By ssumptio B b f f { [, ]: ( ) } ([, ]) is closed i [, b] d does ot coti te poit c Tis mes c is i te complemet of B wic is ope i [, b] Terefore, tere eists > suc tt B(c, ) [, b] Complemet of B i [, b] Tt is, for ll i B(c, ) [, b], f () > Tus by Teorem prt (), f is lower semi-cotiuous t c Tis completes te proof of prt () Proof of Teorem 5 We prove te teorem we f is o-egtive, bouded d Lebesgue itegrble Suppose for ll i [, b], f () < M for some rel umber M Give >, let Te iteger N suc tt N > M b For iteger >, let E b f f { [, ]: ( ) ( ) } ([( ), )} Sice f is Lebesgue itegrble, f is mesurble d E is mesurble Sice E [, b], te mesure of E, m(e ) is fiite It follows tt tere eists ope set G suc tt E G d m( G) m( E) () Let A = G [, b] Te A is ope i [, b] d te crcteristic fuctio 6

7 A is lower semi-cotiuous o [, b] We deduce tis s follows I view of te fct tt { [, b]: ( ) } ([, ]) is eiter empty, ll of [, b] A or te complemet of A i [, b], wic is closed i [, b], it follows from Teorem 6 prt () tt is lower semi-cotiuous o [, b] Let N A u A Te u is lower semi-cotiuous o [, b] sice fiite sum of A lower semi-cotiuous fuctio is lower semi-cotiuous Observe tt for E, Terefore, b N u m( A ) u() > f () ( ) () N N N N m( E ) ( ) m( E ) m( E ) f ( b ) f ( b ) f by (), N b b E (3) Suppose ow f is oegtive but ubouded For ec iteger >, let g () = mi{f (), } Te g is oegtive d bouded d Lebesgue itegrble Defie f = g, d f = g g - for > Plily, ec f is oegtive, bouded d Lebesgue itegrble I prticulr, f f (4) By te first prt of te proof, tt is, iequlity (3) wit replced by, we c fid lower semi-cotiuous fuctio u o [, b] suc tt 7

8 u f o [, b] d b b u f (5) Now let u u Sice fiite sum of lower semi-cotiuous fuctios is lower semi-cotiuous, te -t prtil sum of te series, is lower semicotiuous Moreover, is o-egtive, bouded d coverges poitwise to u (fiite or ifiitely) Te for y rel umber, { [, b]: u( ) } { [, b]: ( ) } is closed i [, b] sice ec { [, b]: ( ) } is closed i [, b] It follows te from Teorem 6, tt u is lower semi-cotiuous I prticulr from (5) we ve, 8 d by usig te Lebesgue Mootoe u u f f Covergece Teorem, u u f f b b b b Filly, suppose f is rbitrry Lebesgue itegrble fuctio o [, b] For ec iteger >, let ow f () = m{ f (), } Plily, f f for ll iteger d f f poitwise o [, b] Terefore, by te Lebesgue Domited Covergece Teorem, b f lim b f Tus, give >, we c coose iteger N so tt b b fn f (6) By defiitio of f N, f N + N So f N + N is oegtive d Lebesgue itegrble Terefore, by wt we ve just proved for oegtive itegrble fuctio, tere is lower semi-cotiuous fuctio u N suc tt N b b u N f N + N d u f N N (7)

9 Now let u = u N N Te u = u N N f N f I prticulr, from (7) we ve, u un N fn b b b b b f f by iequlity (6) To fid upper semi-cotiuous fuctio v for te teorem, we ote tt if fuctio f is lower semi-cotiuous, te f is upper semi-cotiuous d use wt we ve just proved i te followig mer We c fid lower semicotiuous fuctio w for f stisfyig w f d b b w f Now let v = w d so v is upper semi-cotiuous o [, b] Te b b b v w f Tis completes te proof of Teorem 5 Te et ide is to crcterize Lebesgue itegrl i terms of mjor d mior fuctios, ide of Perro wic leds to geerliztio of te Lebesgue itegrl For tis we eed to brig i te ide of upper d lower derivte of fuctio Defiitio 7 Suppose F: [, b] R is rel-vlued fuctio Let c be i [, b] Te te upper derivte of F t c is defied by F( ) F( c) DF( c) limsup c c d te lower derivte of F t c is defied by DF( c) limif c F( ) F( c) c 9

10 It is esy to see tt F is differetible t c if d oly if bot DF( c ) d DF( c ) re fiite d equl We stte some useful results below, strtig wit oe bout cotiuity d te oter bout mootoicity Teorem 8 Suppose F: [, b] R is rel-vlued fuctio Let c be i [, b] If bot DF( c ) d DF( c ) re fiite, te F is cotiuous t c Proof Let M = m DF( c), DF( c ) Sice tere eists > suc tt F( ) F( c) DF( c) limsup, c c F( ) F( c) ( c, c ) { c} DF( c) M, (8) c Similrly, s DF( c) limif c F( ) F( c), tere eists > suc tt c F( ) F( c) ( c, c ) { c} DF( c) M, (9) c Te 3 = mi (, ) Te it follows from (8) d (9) tt F( ) F( c) ( c 3, c 3) { c} M M c F( ) F( c) Tus ( c 3, c 3) { c} M c F( ) F( c) ( M ) c Hece give >, te mi 3, M Te c F( ) F( c) Cosequetly, F must be cotiuous t c

11 Te et teorem is result for sufficiet coditio for fuctio to be icresig Teorem 9 Suppose F: [, b] R is rel-vlued fuctio Suppose DF( ) for ll i [, b] Te F is icresig o [, b] Proof We prove te teorem uder te coditio tt DF( ) for ll i [, b] Let c < d b We sll sow tt F(c) < F(d ) Now DF( c) d so te set H = { [c, d]: F() > F(c)} is o-empty for oterwise, DF( c) would be less t or equl to H is obviously bouded bove by d d so it s supremum M d We clim tt M = d Firstly M must be i H If M does ot belog to H, te it is limit poit of H Sice M is supremum of H, tere eists strictly icresig sequece ( ) i H suc tt M Now F( ) F( M ) F( ) F( M ) lim limif M M F( ) F( M ) limif DF ( M ) M M Terefore, tere eists iteger N suc tt N implies tt F( ) F( M ) DF( M ) M Tus F(M) >F( N ) > F(c) Tis sows tt M is i H Now M must be equl to d If M < d, te sice DF( M ), tere must be poit i te itervl (M, d ) suc tt F() > F(M) for oterwise DF( M ) would be less t or equl to Sice F(M) > F(c), F() > F(c) d so is i H Tis cotrdicts tt M is te supremum of H Hece M = d d F(d) > F(c) Sice c d d re rbitrry, tis sows tt F is strictly icresig o [, b] Suppose ow DF( ) for ll i [, b] Let > Let G() = F() + o [, b] Te DG( ) DF( ) for ll i [, b], sice te derivtive of te fuctio is It follows tt for y d > c i [, b], G(d) > G(c) Tt is, F(d) > F(c) + (d c) Sice we c coose to be rbitrrily smll, F(d) F(c) Tis proves tt F is icresig

12 Te et result below reltes semi-cotiuity wit te upper d lower derivtes Teorem Suppose f : [, b] R* is Lebesgue itegrble eteded rel vlued fuctio Let F( ) f for i [, b] Let c [, b] () If f is lower semi-cotiuous t c, te DF( c) f ( c) (b) If f is upper semi-cotiuous t c, te DF( c) f ( c) Proof () Suppose f is lower semi-cotiuous t c By Teorem prt (), for < f (c), tere eists > suc tt f () > for ll B(c, ) [, b] If f (c) =, we ve otig to prove We ssume tt f (c) > For ll B(c, ) [, b] {c}, F ( ) F ( c ) c f f f c c c c c c Note tt te lst iequlity is obvious if > c If < c, te Tis implies tt DF() c c c f f c c c c for ll < f (c) It follows tt DF( c) f ( c) (b) Suppose f is upper semi-cotiuous t c By Teorem prt (c), for > f (c), tere eists > suc tt f () < for ll B(c, ) [, b] If f (c) =, we ve otig to prove We ssume tt f (c) < For ll B(c, ) [, b] {c}, F ( ) F ( c ) c f f f c c c c c c Tis implies tt DF() c for ll > f (c) It follows tt DF( c) f ( c)

13 Mjor d Mior Fuctios We ow itroduce Perro s mjor d mior fuctios Defiitio Suppose f : [, b] R* is eteded rel vlued fuctio A rel-vlued fuctio U : [, b] R is mjor fuctio of f o [, b], if DU ( ) d DU ( ) f ( ) for ll i [, b] A rel-vlued fuctio V : [, b] R is mior fuctio of f o [, b], if DV ( ) d DV ( ) f ( ) for ll i [, b] Te et result is crcteriztio of Lebesgue itegrble fuctio i terms of mjor d mior fuctios We itroduce te followig ottio If F is rel vlue fuctio we deote F(b) F() by b F Teorem Suppose f : [, b] R* is mesurble eteded rel vlued fuctio Te fuctio f is Lebesgue itegrble o [, b] if d oly if for ec >, tere eist bsolutely cotiuous mjor d mior fuctios, U d V of f o [, b] suc tt U() = V() = d 3 U V Proof Suppose f is Lebesgue itegrble o [, b] Give >, by Teorem 5, tere eist lower semi-cotiuous fuctio u d upper semi-cotiuous fuctio v suc tt for ll i [, b], u() >, u() f () v() d v() <, d u d v re Lebesgue itegrble o [, b] wit u f v b b b () Let U( ) u d V( ) v Te U d V re bsolutely cotiuous fiite fuctios o [, b] By Teorem, DU ( ) u( ) for ll i [, b], sice u is lower semi-cotiuous o [, b] Sice v is upper semi- b b

14 cotiuous o [, b], DV ( ) v( ) Observe tt DU ( ) u( ) f ( ) d DV ( ) v( ) f ( ) for ll i [, b] Hece U is mjor fuctio d V is mior fuctio of f o[, b] Moreover, it follows from () tt b b b b b b b b U V u v u f v f Coversely, give y >, tere eist bsolutely cotiuous mjor d mior fuctios, U d V of f o [, b] suc tt U V Te sice U d V re bsolutely cotiuous, U d V re differetible lmost everywere o [, b] d teir derivtives re Lebesgue itegrble Tt is to sy, te derivtives U d V eist lmost everywere o [, b] d U d V re Lebesgue itegrble Terefore, DU U ' lmost everywere o [, b] d DU is Lebesgue itegrble o [, b] We lso ve b b DV V ' lmost everywere o [, b] d so DV is Lebesgue itegrble o [, b] By defiitio of mjor d mior fuctio of f, DV ( ) f ( ) DU ( ) for ll i [, b] Plily b b b b DU DV U ' V ' U V by bsolute cotiuity of U d V b b Tus we ve sow tt give y >, tere eist itegrble fuctios g d suc tt g f d itegrble o [, b] Tis completes te proof Remr b b g It follows tt f is Lebesgue Suppose U d V re mjor d mior fuctios of f o [, b] Te D( U V )( ) DU ( ) DV ( ) f ( ) f ( ) for ll i [, b] Tus, by Teorem 9, U V is icresig o [, b] Terefore, U b V b If f s t lest oe mjor d mior fuctio, te we defie te upper Perro itegrl o [, b] to be UPf if{ U b : U mjor fuctio of f } d te lower Perro itegrl to be LPf sup{ V b : V mior fuctio of f } 4

15 Te we ve LPf UPf If LPf UPf, te we sy f is Perro itegrble o [, b] Teorem te sys tt y Lebesgue itegrble fuctio o [, b] is Perro itegrble Te et result is id of limit covergece teorem for te Lebesgue itegrl Teorem 3 Suppose f : [, b] R* is Lebesgue itegrble Tere re sequeces of cotiuous fuctios ( p :[, b] R ) d ( P :[, b] R ) suc tt (i) p () = P () =, (ii) p ( ) f P f uiformly o [, b], ( ) d Dp ( ) f ( ) DP ( ), weever f () is fiite Proof Let By Teorem, tere eist mjor fuctio U d mior fuctio V of f o [, b] suc tt DU ( ), DV ( ), DV ( ) f ( ) DU ( ) for ll i [, b] d U( b) U( ) V ( b) V ( ) d U( ) V( ) Moreover we c deduce from te proof of Teorem, tt te mjor d mior fuctios stisfy U( ) f V( ) d U( ) V ( ) U( b) V( b) for ll i [, b] Tus U( ) f U( ) V ( ) for ll i [, b] It follows tt uiformly o [, b] Similrly we deduce tt V ( ) U ( ) f uiformly o [, b] Now let p( ) V( ) d P( ) U( ) Te p ( ) f P f uiformly o [, b] Moreover, d ( ) f Dp ( ) DV ( ) f ( ) DU ( ) DP ( ) Tis completes te proof of te teorem 5

16 Remr Observe tt D( U ( ) V ( )) DU ( ) DV ( )) for ll i [, b] Terefore, by Teorem 9, U ( ) V ( ) is icresig d oegtive i [, b], sice U ( ) V ( ) Te fuctios p () d P () re lso ow s de l Vllée-Poussi s miort d mjort fuctios Sectio B Riem s Ide, Symmetric Secod Derivtive, R-Summbility d Coveity Let A ( ) cos( ) b si( ) for > d A () = We ow write te trigoometric series (A) s T ( ) cos( ) b si( ) A ( ) A ( ) (A) Observe tt A ( ) cos( ) b si( ) b cos( ), were b cos( ) d si( ) b b Let b We derive first ecessry coditio for covergece of te series (A) Teorem 4 If A (), or i prticulr, if T() coverges i set E of positive mesure, te d b Proof It is eoug to prove tt b We prove tis by cotrdictio Suppose Tis mes tere eists > d subsequece of suc tt for ll positive iteger Note tt if T() coverges i set E of positive mesure, te A () i set E of positive mesure Sice A () i E, it follows tt cos( ) i E Sice cos( ) is uiformly bouded by, by te Bouded Covergece Teorem, 6

17 cos ( ) ( ) cos ( ) E d d E But cos ( ) cos( ) d so () E cos ( ) d m( E) cos( ) d E Note tt cos( ) cos( )cos( ) si( )si( ) Terefore, E cos( ) d ---- (), cos( ) cos( ) ( ) d si( ) si( ) ( ) d E E By te Lebesgue Riem Teorem, cos( ) ( ) d d si( ) ( ) d E E d s cos( ) d si( ) re bouded sequeces, it follows from () tt E cos( ) d Tus we obtied from () tt cos ( ) d m( E) E Tis cotrdicts cos ( ) d E Terefore, b d so d b If we formlly itegrte te series (A) term by term twice, we sll obti te followig series A ( ) ( ) (U) 4 7

18 A Let ( ) ( ) Te ( ) ( ) 4 We ve lredy proved tt if T() coverges or A () o set of positive mesure, te d b Cosequetly, A () uiformly o R d (U) gives very useful series We stte it formlly below Teorem 5 If T() coverges i set E of positive mesure, te A ( ) ( ) coverges bsolutely d uiformly to cotiuous fuctio o R, d so ( ) ( ) coverges bsolutely d uiformly to 4 cotiuous fuctio o R Proof Sice te sequece ( A () ) is uiformly bouded by Teorem 4, it A follows by Weierstrss M-test tt ( ) ( ) coverges bsolutely d uiformly to cotiuous fuctio o R Te secod sttemet is ow obvious To pproc te problem of uiqueess, Riem s ide is to rgue bcwrds from ( ) to T() by process of geerlized symmetric secod derivtive Te Ide of Symmetric Secod Derivtive Defiitio 6 Suppose g is fiite fuctio, ie, rel-vlued fuctio For y rel umber, defie ( ) ( ) ( ) ( ) g g g g g ( ) If te limit lim eists, te tis limit is clled te geerlized (symmetric) secod derivtive of g t We deote tis by D g() Tt is, 8

19 g ( ) Dg( ) lim We ow describe some properties of symmetric secod derivtive d some of its vrits Te first step towrds provig Teorem is te followig: Teorem 7 (Riem) Suppose T() is trigoometric series, were d b Let ( ) ( ) s i (U) If T() coverges to f (), 4 te D ( ) f( ) Proof Suppose T() coverges to f () Defie R ( ) ( ) Te limit of R ( ) is D ( ) 4 Observe tt ( ) ( ) ( ) ( ) R ( ) 4 4 si( ) A ( ) (3) fter pplyig te dditio formul d summig We wt to prove tt R( ) T( ) A( ) Let s( ) A( ) for Te s ( ) T( ) We ow itroduce some ottio, write for =, si( ) Let si( ) for, d Let be te prtil sum of R () defie by 9

20 si( ) A( ) (4) Here we let A( ) isted of oly for te proof of tis teorem By Abel s summtio formul, s s s s, (5) were d s =s () Sice s ( ) T( ) d si( ) for ec s, si( ) si(( ) ) s s ( ) for Terefore, for, ( ) si( ) si(( ) ) R( ) s 4 ( ) (6) Sice s ( ) T ( ) s, we my write s = s +e d e Te for fied, R ( ) ( s e ), s s s s ( s e ) s s e e Te for < N <, N s e e e N

21 N e e dt e ( ) d si ( t) N dt t N e e dt e ( ) d si ( t) m m N N dt t (7) sice Let G(t) be te derivtive of, si ( t) t for t We te obti from (7), for N ( ) s e m e G( t) dt m e N N N ( ) e m e G( t) dt m e N N Te by pssge to te limit, we ve, sice m e N N (8) N R ( ) s e m e G( t) dt Note tt for t, d si ( t) tsi( t) si ( t) Gt () so tt for t >, 3 dt t t Gt () tt 3 t t It follows from tis iequlity tt G () t dt Observe lim Gt ( ) Terefore, t G () t dt C It follows from (8) tt G () t dt Ideed G( t) dt Hece, N N R ( ) s e m e G( t) dt e m e C N N

22 Give > we my coose sufficietly lrge N so tt m m e Tus for tis vlue of N we ve N e sice C N R ( ) s e (9) Sice si( ) si(( ) ) lim lim ( ) for ec =,,, N-, N e lim So tere eists > so tt for < <, N e Terefore, it follows from (9) tt for < <, R We c tus coclude tt ( ) s R( ) s T( ) A( ) Observe tt we ve proved more geerl result cocerig R summbility Defiitio 8 I oour of Riem, if series si( ) u u s s, we sy te series u is R- summble to te sum s Tus Teorem 7 sttes tt if te trigoometric series T() stisfies d b d coverges to s t, te it is R- summble to s t We ve ctully proved te regulrity of R-summbility We stte te result s follows

23 Teorem 9 If to te sum s u coverges to te sum s, te te series is R- summble (Te proof of Teorem 9 is lmost ectly te sme s i Teorem 7 ecept for pproprite cge i ottio ) Now we ivestigte some properties of te symmetric secod derivtive d its reltio to coveity Defiitio Let Dg g ( ) ( ) limsup d g ( ) Dg( ) limif If g ( ) o ( ) or equivletly g ( ) lim, te g is sid to be smoot t g is sid to be smoot i set i itervl if it is smoot t every poit i te set Note tt if g is differetible t, te g is smoot t Teorem Suppose g is cotiuous i (, b) d Dg i (, b) ecept perps i eumerble set E If E is empty, te g is cove If E is ot empty d g is smoot i E, te g is cove i (, b) Proof Suppose D g( ) i (, b) ecept for i E Note tt g is cove o (, b) if for y < i (, b), ( ) g( ) ( ) g( ) g ( ) for ll i [, ] g is cocve o (, b) if g is cove o (, b) Suppose o te cotrry tt g is ot cove Te tere is itervl [, ] i (, b) suc tt 3

24 ( ) g( ) ( ) g( ) g ( ) for some i (, ) Tt is to sy, te fuctio d() = g() m, were m g ( ) g ( ) ( ) ( ) d g g is sometime positive Note tt d() = d() = Sice g is cotiuous, d is lso cotiuous Hece by te Etreme Vlue Teorem, tere eists i (, ) suc tt d( ) is te bsolute mimum of d [, ] I prticulr, d( ) > Terefore, for sufficietly smll so tt [, +] (, ) g( ) d( ) d( ) d( ) d( ) It follows tt Dg( ) If te eceptiol set E is empty, te tis would cotrdict Dg( ) If te eceptiol set E is o-empty, te E We sll sow tt E is o-eumerble Let = d( ) > Sice d is cotiuous t tere eists > suc tt for ll i (, + ), d() > d( ) /4 = 3/4 > Terefore, for i (, + ), g( ) ( m ) d( ) Let (m(, ) ) 3 4 If (m(, ) ), te for ll i [, ] (m(, ) ) d so for i (, + ), 3 g( ) ( m ) d( ) () 4 4 Tis mes tt for ll sufficietly smll er m, ie, for m <, 4

25 g( ) for i (, + ) Let d ( ) g( ) Te d lso s positive bsolute mimum i [, ] Observe tt d( ) g( ) m ( m ) d( ) ( m ) Hece d( ) d( ) ( m ) ( m ) ( m) d d( ) d( ) ( m ) ( m ) ( m) Let be te bsolute mimizer of d i [, ] Observe tt d( ) d( ) d( ) ( m ) ( m ) ( m ) d( ) d similrly tt d ( ) ( m ) d ( ) Hece (, ) Let { [, ]: d ( ) d ( )} Te is o-empty d is cotied i [, ] Let sup Te [, ] Sice d ( ) d ( ), d ( ), (, ) d d ( ) d ( ) It follows s i te cse for tt Dg Dd ( ) ( ) Hece E We sll sow et tt g is differetible t d tt g( ) Note tt is mimizer of d i [, ], we ve te for sufficietly smll positive, d ( ) d ( ) d d ( ) d ( ) It follows tt te rigt upper derivte of d t, D d D d d ( ) d ( ) ( ) limsup d te lower left derivte d ( ) d ( ) ( ) limif Sice g is smoot i E, d is lso smoot i E, ie, for i E, or d ( ) o ( ), 5

26 d( ) d( ) d( ) d( ) lim lim () Tus d( ) d( ) d( ) d( ) lim lim d so give y > tere eists > so tt < < implies d ( ) d ( ) d( ) d( ) d( ) d ( ) d ( ) < Tis mes, for < < d ( ) d ( ) d ( ) d ( ) d ( ) d ( ) It follows tt d ( ) d ( ) d ( ) d ( ) limsup limsup d d ( ) d ( ) d ( ) d ( ) limsup limsup Sice is rbitrry, d ( ) d ( ) d ( ) d ( ) D d ( ) limsup limsup D d( ) Hece Dd ( ) D d ( ) Similrly by usig limit iferior we c sow tt Dd ( ) D d ( ) Terefore, Dd ( ) Dd ( ) d so Dd ( ) Dd ( ) d d is differetible t wit d ( ) 6

27 By defiitio of d, d ( ) g( ) d so g( ) Tus for ec i te itervl, (m, m+ ), we c ssocite elemet i E suc tt g( ) Terefore, tere re s my elemets i E s tere re i (m, m+ ) Tis mes tt E cotis set wic is o-deumerble d so E is o-deumerble set Tis cotrdicts tt E is deumerble It follows tt g must be cove Teorem Suppose g is cotiuous i (, b) d Dg i (, b) ecept perps i eumerble set E If E is empty, te g is cove i (, b) If E is o-empty d g is smoot i E, te g is cove i (, b) Proof For ec iteger >, let g ( ) g( ) Te Dg ( ) Dg ( ) for ll ecept for i E If g is smoot i E, g is lso smoot i E Terefore, by Teorem, ec g is cove i (, b) Sice g is te limit of g, g is lso cove i (, b) For cocvity we ve te followig result Teorem 3 Suppose g is cotiuous i (, b) d Dg i (, b) ecept perps i eumerble set E If E is empty, te g is cocve i (, b) If E is o-empty d g is smoot, te g is cocve i (, b) Proof Let = g Te D D( g) Dg Note tt is cotiuous i (, b) d smoot i E if E is o-empty, sice g is Terefore, by Teorem, is cove d so g is cocve i (, b) Corollry 4 Suppose g is cotiuous i (, b) d Dg i (, b) ecept perps i eumerble set E If E is empty, te g is lier i (, b) If E is o-empty d g is smoot i E, te g is lier i (, b) Proof If Dg, te Dg Dg Te by Teorem d Teorem 3 g is bot cocve d cove i (, b) Terefore, for y itervl [, ] i (, b) g is lier fuctio o [, ] d te derivtive g is costt i [, ] 7

28 By lettig teds to d teds to b, we coclude tt g is costt fuctio i (, b) It follows tt g is lier fuctio Teorem 5 Suppose g is cotiuous i (, b) d Dg c i (, b) ecept perps i eumerble set E i wic g is smoot if E is o-empty Te g ( ) c for < < + < b; Suppose g is cotiuous i (, b) d Dg c i (, b) ecept perps i eumerble set E i wic g is smoot if E is o-empty Te < < + < b g ( ) c for Proof Let p( ) g( ) c E if E is o-empty Te Dp Dg c i (, b) ecept perps i E Te p is cotiuous i (, b) d smoot i Terefore, by Teorem, p is cove i (, b) Sice p is cove i (, b), for y d suc tt < < + < b, p( ) g( ) c d so < + < b Similrly, if D g( ) ( ) But p g g ( ) c for < c i (, b) ecept perps i ( ) c Hece eumerble set E i wic g is smoot, te D( g) Dg c d so by wt we ve just proved, for < < + < b, d it follows tt g ( ) c ( g)( ) g( ) c Our et result is bout smootess of te fuctio () obtied by forml itegrtio of trigoometric series T() twice To get bc to te trigoometric series usig te symmetric secod derivtive we eed to use te smootess of () 8

29 Teorem 6 Suppose T() is trigoometric series tt coverges i set E of positive mesure or d b Te te fuctio A ( ) ( ) 4 obtied by formlly itegrtig te trigoometric series T() twice, is cotiuous d smoot o te wole of R Tt is to sy, ( ) s or ( ) o( ) Proof We sll sow tt (3), ( ) Te teorem te follows From ( ) si( ) 4 R( ) A ( ) We sll write te summtio i tree prts Sice A ( ) uiformly o R, give >, tere eists iteger N suc tt > N implies tt A ( ) i R Te first prt is N si( ) ( ) I A for ll Te secod prt is cose ccordig to Give y, wit <, let s be te iteger prt of Te we ve s s Tus s ( s ) Prt (II) is give by II Ns si( ) A ( ) N, Prt (III) is give by 9

30 III si( ) A ( ) N s Te we ve ( ) I II III () 4 Observe tt for y, I s sice si( ) Sice A ( ) for ll d for ll > N, N s N s d so II A ( ) s N N II s (3) Observe tt III A ( ) N s N s N s sice d ( s ) s s for s Terefore, sice ( s N) ( s ), III Hece III It follows te usig (3) d (4) tt (4) ( ) ( ) I II III I 4 Sice I s, tere eists > so tt for < <, we ve I < Hece for < <, ( ) ( ) 3

31 Sice is rbitrrily cose, tis sows tt ( ) lim Cosequetly ( ) lim for y Tis completes te proof, Icidetlly, we ve proved te followig teorem Teorem 7 If (u ) is sequece tt coverges to, te si( ) u s Te et result is tecicl result ims t epressig te differece of ctully itegrtig fuctio f twice d te fuctio obtied by formlly itegrtig te Fourier series of f twice term by term Teorem 8 Suppose f () is fiite ecept i eumerble set E d itegrble i (, b) Suppose g() is cotiuous i (, b) d smoot i E we E is o-empty d tt D g f D g ( ) ( ) ( ) t for ll i (, b) ot i E Let J ( ) ( ) of f Te g() J() is lier i (, b) f u du dt be te repeted itegrl Proof We sll employ de L Vllee Poussi mjort d miort fuctios Sice f is Lebesgue itegrble i (, b), by Teorem 3, tere re sequeces of cotiuous fuctios ( p :[, b] R ) d ( P :[, b] R ) suc tt (i) p () = P () =, (ii) p ( ) f i E,, P ( ) Dp ( ) f ( ) DP ( ) 3 f uiformly i [, b] d for ot

32 , Let q ( ) p ( t) dt d Q ( ) P ( t) dt Sice p ( ) f P ( ) f uiformly i [, b], q ( ) J ( ) d Q ( ) J ( ) uiformly i [, b] Note tt bot q ( ) d Q ( ) re differetible i [, b] By te Cucy Me Vlue Teorem, Q ( ) Q ( ) Q ( ) Q( ) Q ( ) Q ( ) for some betwee d Tus tere eists < < suc tt Hece, Q ( ) Q ( ) Q ( ) P ( ) P ( ) P ( ) P ( ) P ( ) P ( ) (5) Terefore, it follows from (5) tt Q ( ) P ( ) P ( ) P ( ) P ( ) limif limif limif Tis mes DP ( ) DP ( ) DP ( ) ecept for i E, sice DP ( ) f ( ) D Q( ) DP( ) f ( ), (6) Let ()= J() g(), K () = Q () g() d () = q () g() Te K () () d () () uiformly i (, b) Note tt sice J(), q () d Q () re differetible d g is cotiuous i (, b), (), K () d () re ll cotiuous i (, b) If E is o-empty, sice g is smoot i E, (), K () d () re ll smoot i E To proceed furter we use te followig iequlity for supremum d ifimum: 3

33 For y two fuctios u() d v() cotiuous i (, ) { } for some >, if { u( ) v( )} sup { u( )} if { v( )} (, ) { } (, ) { } (, ) { } sup { u( ) v( )} (, ) { } provided sup { u( )} if { v( )} is ot of te form - (, ) { } (, ) { } so tt, limif{ u( ) v( )} limsup{ u( )} limif{ v( )} limsup{ u( ) v( )} Te usig (7), ecept for i E, (7) D K ( ) D Q ( ) D ( g( )) D Q ( ) D g( ) f ( ) f ( ) Terefore, by Teorem, K () is cove i (, b) Hece () beig te limit of K () is lso cove i (, b) Similrly we c deduce tt for some < < q ( ) q( ) q( ) q( ) q( ) From tis we c deduce s bove for Q () tt Terefore, for ot i E, D q ( ) D p ( ) f ( ) (7) D ( ) D ( ) D q ( ) D ( g)( ) D q ( ) D g( ) f ( ) f ( ) Terefore, by Teorem 3, () is cocve i (, b) Hece () beig te limit of () is lso cocve i (, b) Tus ()= J() g()) is bot cove d cocve i (, b) d so is lier i (, b) Tis completes te proof 33

34 Itegrtig A Fourier Series Formlly Suppose T() is te Fourier series of Lebesgue itegrble fuctio f Te by te Riemm Lebesgue Teorem, its Fourier coefficiets, d b We sll sow tt by formlly itegrtig te Fourier series term by term, we sll obti uiformly coverget series covergig to te itegrl of f Te Fourier series eed ot coverge d te series so obtied is lwys uiformly coverget Te specil series si( ) S ( ) plys role i tis ivestigtio Note tt S() coverges uiformly i y closed itervl free from multiples of d coverges boudedly to te fuctio J() defied by, ( ),, J( ), d eteded to wole of R by periodicity Tt it coverges boudedly is by Teorem 4 of my ote o Fourier cosie d sie series Tt S() coverges uiformly i y closed itervl free from multiples of my be deduced by usig Diriclet s Test See Teorem 9 i Fourier cosie d sie series Suppose T ( ) cos( ) b si( ) A ( ) A ( ), is te Fourier series of te Lebesgue itegrble fuctio f Cosider te series obtied by formlly itegrtig te series term by term, tt is, cos( ) b si( ) A ( ) cos( t) b si( t) dt A ( t) dt si( ) b cos( ) b b b cos( ) si( ) 34

35 Let B ( ) b cos( ) si( ) for iteger Te te bove series is give by b B ( ) ( W ) (8) We sll sow tt tis series W() is uiformly coverget d coverges to te fuctio F( ) f ( t) dt By defiitio of te Fourier coefficiet b, bm si( m) f ( )si( m) d f ( ) d m m m m m m f ( ) J ( ) d f ( ) ( ) d by te Lebesgue Domited Covergece Teorem s deduced below Sice si( ) S( ) J ( ) boudedly, umber K d for ll iteger Terefore, si( m) K for some rel m m si( m) f ( ) K f ( ) m d sice K f is Lebesgue itegrble, we ivoe te Lebesgue Domited Covergece Teorem to give te bove sttemet Hece we ve lwys coverget d b f ( ) ( ) d m b is Hece we ve proved te followig teorem Teorem 9 If te fuctio f is Lebesgue itegrble d represeted by te T ( ) cos( ) b si( ) A ( ) A ( ) Fourier series is coverget d, te b f ( ) ( ) d b 35

36 Cosider ow F( ) f ( t) dt f ( t) dt o [, ] Note tt F() = F() = Observe tt F is bsolutely cotiuous o [, ] d periodic o R wit period Te si( ) si( ) F( )cos( ) d F( ) f ( ) d by itegrtio by prts, si( ) si( ) ( ) f d d b d cos( ) cos( ) F( )si( ) d F( ) f ( ) d cos( ) cos( ) ( ), f d d by itegrtio by prts Hece te Fourier coefficiets for F() re ( b, ) for ( ) ( ) ( ) ( ) F d F t t F t tdt f t tdt by itegrtio by prts ( ) 4 f ( t ) tdt tdt f ( t ) tdt ( ) b by Teorem 9 f ( t) tdt f ( t) dt t f ( t) dt 4 Hece te Fourier series of F() is give by b b cos( ) si( ) b B( ) W( ) We my ivoe te teory of Fourier series of cotiuous fuctio of bouded vritio tt its Fourier series coverges to te fuctio 36

37 For ow we sll sow its covergece directly by simple device of siftig te fuctio itself Fi i [, ] Let g(t) = f (+t) Here f is defied outside [, ] by periodicity Te g is Lebesgue itegrble sice f is We sll cosider te Fourier coefficiets of g(t) g( t)cos( t) dt f ( t )cos( t) dt f ( u)cos( ( u )) du d cos( ) f ( u)cos( u) du si( ) f ( u)si( u) du cos( ) b si( ) A ( ) g( t)si( t) dt f ( t )si( t) dt f ( u)si( ( u )) du cos( ) f ( u)si( u) du si( ) f ( u)cos( u) du b cos( ) si( ) B ( ) Terefore, te Fourier coefficiets of g(t) is give by ( A ( ), B ( )) It follows te by Teorem 9 tt B ( ) is coverget d coverges to g( t) ( t) dt f ( t ) ( t) dt F ( t ) ( t) dt sice ( t) dt F( t )( t) F( t ) dt F( )( ) F( ) F( t ) dt F( ) F( t ) dt (9) Now 37

38 ( ) ( ) ( ) F t dt F u du F u du by periodicity ( ) ( ) ( ) F u u F u udu f u udu u f ( u) du sice udu f ( u) du by Teorem 9 b Terefore, it follows from (9), g( t) ( t) dt f ( t ) ( t) dt F( ) Hece B ( ) f () t dt f () t dt b b b Terefore, te series b b cos( ) si( ) b B ( ) W( ) f t dt F coverges to ( ) ( ) We ve tus proved te covergece prt of te followig teorem Teorem 3 Suppose f is periodic wit period d is Lebesgue itegrble Its Fourier series my be itegrted term by term d te itegrted series coverges uiformly d b b cos( ) si( ) b B ( ) f () t dt, ie, te rigt d series coverges uiformly to f () t dt 38

39 Proof We ve lredy proved covergece We ow sow tt te covergece is uiform It is sufficiet to sow tt te covergece of B ( ) is uiform i Recll tt te Fourier coefficiets of g(t) = f (t +) is give by ( A ( ), B ( )) Terefore, q p B ( ) q si( t) f ( t) dt (3) p Note tt for y p, p si( t) K (3) (See (8) of Fourier Cosie d Sie Series) q ( ) q si( ) q B t si( t) (3) f ( t) dt f ( t) dt p p p si( t) si( t) si( t) f ( t) dt f ( t) dt f ( t) dt q q q p p p q si( t) f ( t) dt p si( t) K f t dt K f t dt f t dt q ( ) ( ) ( ) p (33) Give y >, by te bsolute cotiuity of te Lebesgue itegrl o itervl, tere eists > suc tt for y mesurble subset E of mesure less t, f Tus te y < <, we ve from (3) d (33) E 8K tt q p B ( ) q si( t) (34) f ( t) dt p Sice si( t) coverges uiformly o te itervl [, ], it is uiformly Cucy d so tere eists iteger N suc tt for ll itegers q p N d for ll t i [, ], 39

40 q p si( t) f ( t) dt) (35) Tus it follows from (34) d (35) tt for y, d for q p N, q p B ( ) f ( t) dt f ( t) dt) ( ) ( ) f t dt f t dt ( ) ) ( ) ) f t dt f t dt Tis sows tt B ( ) is uiformly Cucy o R d so uiformly o R Tis completes te proof of Teorem 3 B ( ) coverges Now we my use Teorem 3 to ivestigte te reltio of te symmetric secod derivtive wit te double itegrl of f Suppose f is periodic d itegrble o fiite itervl d T() is its Fourier series Te usig itegrtio by prts we ve Let (, t) f ( t) f ( t) (, t)( t) dt (, t) dt s (, t) dt ds s s (36) I view of Teorem 3, we c epress te itegrl o te rigt side s series Teorem 3 Suppose f is periodic d Lebesgue itegrble Let ( ) si( ) 4 R( ) A ( ) s defied i Teorem 7 usig te Fourier series T() of f Te si( ) R( ) A ( ) (, t)( t) dt 4

41 Proof Usig (36) we obti (, t)( t) dt (, t) dt s (, t) dt ds s s s (, t) dt ds (37) (, t) dt f ( t) f ( t) dt f ( t) dt f ( t) dt s s s s Now s s f ( t) dt f ( t) dt (38) Te Fourier series for g(t) = f (+t) is give by ( A ( ), B ( )) I,e, its Fourier series is A B ( )cos( ) ( )si( ) Terefore, by Teorem 3, B ( ) B ( )cos( ) A ( )si( ) f ( t ) dt (39) d te series o te rigt d side coverges uiformly i Hece we ve B ( ) B ( )cos( ) A ( )si( ) f ( t ) dt (4) It follows te from (38), (39) d (4) tt A ( )si( ) (, t) dt, (4) d te series o te rigt d side of (4) coverges uiformly i for i R Terefore, we c itegrte (4) term by term, obtiig s (, t) dtds A ( ) A ( )cos( ) 4 4

42 Hece usig (37), d (4), It follows tt si (4) 4 A ( ) si ( ) (, t)( t) dt A ( ) si( ) (, t)( t) dt A ( ) R( ) Tis proves Teorem 3 By Teorem 9, if A ( ) coverges to vlue c, te te series is R summble to c, ie, R () c s Wit Teorem 3 we my ve differet wy of determiig R summbility by usig te itegrl (, t)( t) dt From (4) we obti A ( )si( t) t t (, u) du (43) t Now if f is Lebesgue itegrble, te te fuctio F( ) f ( t) dt is bsolutely cotiuous, differetible lmost everywere d F( ) f ( ) lmost everywere Hece for lmost ll, t f ( u) du f ( u) du d lim f ( ) (44) t t t lim f ( ) t t 4

43 By usig te fuctios g(t) = f (+t) d (t) = f (t) wit G( ) g( t) dt d H( ) ( t) dt, te bove is just te sttemet G() g() f ( ) d H() () f ( ) It follows from (44) tt for lmost ll, t lim (, u) du f ( ) t t (45) Teorem 3 Suppose f is periodic d Lebesgue itegrble Te for lmost ll, lim (, t)( t) dt f ( ), tt is, for lmost ll, R () f ( ) Ideed if t lim (, u) du c t t, te lim (, t)( t) dt c or R () c Proof I view of (45) it is sufficiet to prove tt implies lim (, t)( t) dt c t lim (, u) du c t t (, t)( t) dt (, t) dt (, t) tdt c lim (, t) tdt (46) (, t) tdt (, t) c tdt ctdt (, t) ctdt c (47) 43

44 lim (, ) c tdt We clim tt t s s (, ) (, ) (, ) s (, t) cdt (, t) cdtds t c tdt t c dt s t c dt ds s (, t) dt c (, t) cdt ds Now (48) (, t) dt c c c We sll sow tt s (, t) c dt ds s Sice t lim (, ) c dt s s, give >, tere eists, > suc tt for s s s < s <, (, t) cdt or (, ) t c dt s (49) Terefore, for, for >, s s (, t) c dt ds (, t) c dt ds sds d for <, (, ) (, ) s s t c dt ds t c dt ds s ds s Tus for, (, t) cdt ds s (, t) cdt ds (, t)( t) dt c lim (, t) tdt c c c It follows tt Terefore, from (46), (47) d (48) we get Tus R () c 44

45 As cosequece of Teorem 3, we ve: Teorem 33 Suppose f is periodic d Lebesgue itegrble Te for lmost ll, its Fourier series T() is R-summble to f () We ve ctully proved te followig: Teorem 33A If te Fourier series of f is Lebesgue summble t to c, te it is Riem summble or R-summble to c (For te defiitio of Lebesgue summbility, see pge 4 of Abel-summbility of Fourier series d its derived series d Teorem tere) Sectio C Uiqueess Teorems Uiqueess of Fourier d Trigoometric Series Our first uiqueess teorem cocers trigoometric series Teorem 34 If two trigoometric series coverge to te sme sum ecept i eumerble set E, te tey re ideticl More precisely if trigoometric series T() coverges to ecept i E, te = d b = for ll, ie, T() is ideticlly Proof Suppose T() coverges to ecept i E Te by Teorem 4, A d b It follows te by Teorem 5, ( ) ( ) coverges bsolutely d uiformly to cotiuous fuctio o R, d ( ) ( ) 4 is cotiuous o R By Teorem 6, ( ) is smoot o te wole of R By Teorem 7, ecept for i E, D ( ) It te follows from Corollry 4 tt ( ) is lier fuctio o R 45 A Suppose ( ) m C Observe tt ( ) ( ) is bouded fuctio o R sice d b Terefore, ( ) K for some K d for ll i R

46 Tus ( ) ( ) ( ) K We clim tt = If, te by te bove iequlity, sice K s +, 4 ( ) s But ( ) C m m s Tis sows tt = Hece ( ) ( ) m C Sice ( ) is bouded, m = Tus ( ) is te A costt fuctio C, ie, ( ) C for ll i R Tus ( ) ( ) coverges uiformly to te costt fuctio C We my tus itegrte ( ) term by term, givig A ( ) d C ( C) d ( ) d, sice A ( ) d cos( ) d b si( ) d Hece C = d so ( ) for ll i R Agi, sice covergece of ( ) to is uiform, ( )cos( ) d d b ( )si( ) d Terefore, = b = for ll We ve lredy sow tt = Hece te trigoometric series is ideticlly Our et result pves te wy for te proof of Teorem It sttes tt Teorem is true if te trigoometric series coverges ecept i eumerble set to bouded fuctio Teorem 35 Suppose te trigoometric series T() coverges ecept i eumerble set E to bouded fuctio f (), te it is te Fourier series of f () 46

47 Proof Sice T() coverges to f () ecept for i E of zero mesure, T() coverges to f () lmost everywere d so f is mesurble Sice f is lso bouded, f is Lebesgue itegrble o y bouded itervl d i prticulr o [, ] By Teorem 7, ecept for i E, D ( ) f( ) Suppose f () M for some rel umber M d for ll ot i E Te D ( ) f ( ) M ecept for i eumerble set E Terefore, D( ) D( ) M ecept for i E By Teorem 6, is cotiuous d smoot o te wole of R Terefore, by Teorem 5, ( ) M for ll Similrly, sice D( ) D ( ) M, by Teorem 5, ( ) M for ll i R It follows tt ll Tis mes tt R ( ) M for ll d ( ) ( ) is uiformly bouded i d ll 4 si( ) Recll tt R( ) A ( ) Lebesgue Bouded Covergece Teorem, R ( )cos( ) d si( ) cos( ) d A ( )cos( ) d Terefore, by te si( ) s d R ( )si( ) d si( ) si( ) d A ( )si( ) d si( ) b b s 47

48 Sice R ( ) f( ) boudedly lmost everywere R ( )cos( ) f ( )cos( ) boudedly lmost everywere d so by te Lebesgue Bouded Covergece Teorem, R ( )cos( ) d f ( )cos( ) d Tis sows tt f ( )cos( ) d is te Fourier coefficiet of f Similrly sice R ( )si( ) f ( )si( ) boudedly lmost everywere d so by te Lebesgue Bouded Covergece Teorem, R ( )si( ) d f ( )si( ) d Terefore, b f ( )si( ) d is te Fourier coefficiet of f Hece T() is te Fourier series of f Now we remove te boudedess coditio o f i Teorem 35 Tis gives te sttemet i Teorem We restte te teorem ere Teorem If te trigoometric series T() coverges ecept i eumerble set E to fiite d itegrble fuctio f, te it is te Fourier series of f T ( ) cos( ) b si( ) A ( ) A ( ) Proof Suppose Let A ( ) ( ) For te proof we sll compre wit te 4 iterted itegrl of f By Teorem 7, ecept for i E, D ( ) f( ) By Teorem 6, is t cotiuous d smoot o te wole of R Let J ( ) ( ) f u du dt be te repeted itegrl of f Te by Teorem 8, () J() is lier i R 48

49 Te fuctio f is Lebesgue itegrble d so it s Fourier series give by Let g( ) A( ) cos( ) si( ) cos( ) si( ) 4 deduce tis s follows Now J( ) f ( t) dt cos( ) si( ) by Teorem 3 Te J() g() is lier i R We g( ) sice te differetited series is uiformly coverget Terefore, J() g() is lier i R Tus sice () J() is lier i R, () g() is lier i R Tt is to sy, ( )cos( ) ( b )si( ) L( ) ( ) 4 is lier i R Suppose L() = A C (5) Sice ( )cos( ) ( b )si( ) is uiformly bouded, s i te proof of Teorem 34, ( ) = d so = Tus A C ( )cos( ) ( b )si( ) (5) Sice te rigt d side of (5) is bouded i R, A = Hece C ( )cos( ) ( b )si( ) Tus te rigt d side coverges uiformly to costt fuctio Te 49

50 ( )cos( ) ( b )si( ) C Cd d d so C = It follows tt ( )cos( ) ( b )si( ) By Teorem 34, ( ) d ( b ) d so d b Tus T() is te Fourier series of te limitig fuctio f () Tis completes te proof Jmes P itegrl d Coverget Trigoometric series We ow cosider te questio of recoverig te coefficiets of coverget trigoometric series Suppose te trigoometric series (A) coverges to fiite fuctio f How c we recover te coefficiets, b? If f is Lebesgue itegrble, te by Teorem, te series is te Fourier series of f d te coefficiets re give by te Euler formul, f ( )cos( ) d, =,,,, b f ( )si( ) d, =,, Suppose f is ot Lebesgue itegrble, ow c te coefficiets, b be determied? Tis questio s bee settled lbeit by differet metods d te ivetio of differet types of itegrtio teory, ivolvig cge of te form of te bove Fourier formul for te coefficiets, by Dejoy, Verblumsy, Mrciiewicz d Zygmud, Burill d Jmes i te period We sll describe oe solutio by Jmes usig is Perro secod itegrl, wic ws med i oour of Perro s metod of defiig Perro itegrl of fuctio Tis is lso clled te P itegrl Tis is iveted to tcle tis problem d te first crucil result is te followig teorem wic we stte before we give descriptio of te P itegrl 5

51 Teorem 36 If te trigoometric series (A) coverges to fiite fuctio f or equivletly, if f is te poitwise limit of coverget trigoometric series i (, ), te f is ecessrily P itegrble Suppose f is defied i itervl (, b) We describe Jmes mjor d mior fuctios of f i lmost te sme fsio s Perro s mjor d mior fuctios ecept we use te geerlized symmetric secod derivtives d require te fuctios to vis t te ed poits Defiitio 37 Suppose f : [, b] R is fiite vlued fuctio Te pir of rel-vlued fuctios M : [, b] R d m : [, b] R re clled respectively te J-mjor d J-mior fuctios of f o [, b], if () M d m re cotiuous o [, b], M() = M(b) = m() = m(b)=, () D M ( ) f ( ) Dm( ) for i (, b) ecept for deumerble set E i (, b), (3) D M ( ), D m( ) for ll i (, b) ecept for deumerble set E i (, b), (4) M() d m() re smoot for ll i E Defiitio 38 Te rel vlued fuctio f : [, b] R is sid to be P itegrble over (,, b), were < < b, if for y >, tere eists pir of J-mjor d J-mior fuctios of f, M d m o [, b], suc tt m() M() We deote by J( ) te commo vlues sup{ m( ): m J-mior fuctio of f }=if{ M( ): M J-mjor fuctios of f } d defie te P itegrl of f to be J(), tt is to sy, f ( ) d J ( ) (,, b) 5

52 Note tt i te defiitio we ve te ito ccout tt if M d m re J- mjor d J-mior fuctios of f, te M m is cove o [, b] Coditio () implies tt D( M m)( ) D M ( ) Dm( ) f ( ) f ( ) for i (, b) ecept for deumerble set so tt by Teorem, M m is cove Te by cotiuity d te fct tt ( M m)( b) ( M m)( ), M() m() d so m() M() We my defie f to be P itegrble over (,, b) if sup{ m( ): m J-mior fuctio of f }=if{ M( ): M J-mjor fuctios of f } d deote te egtive of te commo vlue by f () t dt (,, b) Remr Tt coditio () of Defiitio 37 my be replced by weer coditio tt te iequlity be stisfied ecept for set of mesure zero is sow by Jmes usig o-egtive, icresig d bsolutely cotiuous fuctio give i Lemm, 8, pge 369 of Teory of fuctios by Titcmrs, E C 93, togeter wit te fct tt te idefiite itegrl of bouded icresig fuctio is cove Teorem 39 (Jmes) Te rel vlued fuctio f : [, b] R is P itegrble over (,, b), were < < b, if, d oly if for y >, tere eists pir of fuctios M d m o [, b], stisfyig () M d m re cotiuous o [, b], M() = M(b) = m() = m(b)=, () D M ( ) f ( ) Dm( ) for i (, b) ecept for set E of mesure zero i (, b), (3) D M ( ), D m( ) for ll i (, b) ecept for deumerble set E i (, b), (4) M() d m() re smoot for ll i E, d (5) m() M() We sll give proof of tis teorem lter For te bouds of te vlue of te P itegrl over (,, b), we ve 5

53 Corollry 4 Suppose f : [, b] R is P itegrble over (,, b), were < < b Te for y pir of fuctios M d m stisfyig coditios () to (5) of Teorem 39, we ve M( ) f () t dt m() (,, b) Teorem 4 If f : [, b] R is P itegrble over (,, b) d f = g lmost everywere i (, b), te g is lso P itegrble over (,, b) d f ( ) d g( ) d (,, b) (,, b) Proof By Teorem 39, te fuctio M d m stisfyig coditios () to (5) of Teorem 39 for te fuctio f lso wor for g, sice f = g lmost everywere i (, b) It follows te from Teorem 39 tt g is lso P itegrble over (,, b) Moreover, by Corollry 4, bot itegrls lie i te itervl ( M(), m() ) of legt Sice is rbitrry, te itegrls must be te sme Te et teorem gives descriptive defiitio of te P itegrl Teorem 4 Suppose tt F() is cotiuous i [, b] d tt D F( ) is defied for ll i (, b) ecept for set E of mesure zero, d tt D F( ) d D F( ) re fiite for ll i (, b) wit te possible eceptio of deumerble set E, were F() is smoot If f ( ) DF ( ), were D F( ) is defied d f () = elsewere, te f () is P itegrble over (,, b) for y i (, b) d b f ( ) d F( ) F( ) F( b) (5) (,, b) b b b Proof Let M ( ) m( ) F( ) F( ) F( b) Te b b D M ( ) d D m( ) f ( ) for ll i (, b) ecept for set of mesure zero d Dm( ) d DM ( ) re fiite wit te eceptio of deumerble set were M d m re smoot It follows tt coditios () to (5) of Teorem 39 re stisfied By Corollry 4, 53

54 (,, b) b f ( ) d M ( ) m( ) F( ) F( ) F( b) b b Teorem 43 Suppose F() d G() re two fuctios stisfyig te ypotesis of Teorem 4 Suppose DF ( ) DG( ) for lmost ll i (, b) Te for y i (, b), b b F( ) F( ) F( b) G( ) G( ) G( b) b b b b I prticulr, F() G() is lier fuctio i (, b) Proof Suppose DF ( ) DG( ) for ll ot i E d E is set of mesure zero i (, b) Let D F( ), (, b) E f( ) d, E 54 D G( ), (, b) E g ( ), E Te f () = g() for ot i E Tt is, f = g lmost everywere i (, b) By Teorem 4, f () is P itegrble over (,, b) for y i (, b) d b f ( t) dt F( ) F( ) F( b) (,, b) b b Also by Teorem 4, g () is P itegrble over (,, b) for y i (, b) d b g( t) dt G( ) G( ) G( b) (,, b) b b Sice f = g lmost everywere i (, b), by Teorem 4, tese two P itegrls re te sme, ie, b b F( ) F( ) F( b) G( ) G( ) G( b) b b b b Hece, b b F( ) G( ) F( ) F( b) G( ) G( b) b b b b

55 is lier fuctio Suppose ow te trigoometric series T ( ) cos( ) b si( ) A ( ) coverges everywere to fuctio f Te by Teorem 5, A ( ) cos( ) b si( ) ( ) coverges 4 4 bsolutely d uiformly to cotiuous fuctio o R By Riem Teorem (Teorem 7), D ( ) f ( ) everywere It follows te from Teorem 4 tt f is P itegrble over (,, ) for i (, ) d tt (,, ) f ( t) dt ( ) ( ) ( ) 4 4 I prticulr, (,, ) f ( t) dt () ( ) ( ) ( ) ( ) 4 4 Cosequetly, Hece we ve, f () t dt (,, ) Teorem 44 If te trigoometric series T ( ) cos( ) b si( ) coverges everywere to fuctio f, te f is ecessrily P itegrble over (, c, ) for y c i te itervl (, ) d f () t dt (,, ) Remr Teorem 36 of course follows from Teorem 44 55

56 Teorem 45 If te trigoometric series T ( ) cos( ) b si( ) coverges everywere to fuctio f, te for >, f ( t)cos( t) dt d b ( )sios( ) f t t dt (,, ) (,, ) We sll write f () cos() s te limit of trigoometric series We employ te followig tecique s eplied by R L Jefferey i is 953 lecture o Trigoometric Series to te Royl Society of Cd Lemm 46 Suppose K m, K, Km re comple umbers suc tt Km + K + Km Let, c, c,, c, c,, c, be sequece of rel umbers wit c, c s Te for ec m >, K mc m Kc Kmc m lim for ec positive iteger Proof Let S K c K c K c Te m m m m S K c K c K c m m m m K c K c K c m m m m m m m m K K K c K c K c K c K c m m m m m m m m m m m K c K c c K c m m m m m m m K c K c c K c m m m m m m 56