CAPITOLUL IV SERII FOURIER
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1 CAPITOLUL IV SERII FOURIER Sr Fourr ptru uţ Fuţ prod Trsormt prodă Dzvotr î sr Fourr u uţ prod u prod Empu Fuţ prod osttu u d s d uţ r dtortă proprtăţor or trv rvt î dvrs prom tort ş prt U mjo d rprztr ş studu stor uţ î osttu dzvotr î sr Fourr Î mut zur dzvotr î sr Fourr st m ovă dât dzvotr î sr Tyor Trm u sr Fourr sut uţ prod u r putm dsr om ostor O tă tt sror Fourr st ş ă trm să u proprtt d ortogott Spum ă uţ : R Γ Γ R C) st o uţ prodă d prodă T > dă: + T ) ), R Dă T st prod uţ ) tu ş T, Z, * st prodă F supp [,] Numm trsormt prodă uţ, uţ ω R Γ, dtă pr rţ ~ T ω, Trsormt prodă ω ) st o ) ) + T ), R T uţ prodă d prodă T Dţ Pr poom trgoomtr d ordu îţgm uţ: ) T ) + os + s ) ud oţ,, {,,, }) sut umr r Osrvăm ă poomu T ) d ) st o uţ prodă d prodă T Dţ Numm sr trgoomtră sr d orm: ) os s ) + + Dă sr trgoomtră ) st ovrgtă, tu sum ) v o uţ prodă d prodă T Sr trgoomtră s- oţut u jutoru sstmuu trgoomtr udmt : T : 3),os,s,os,s,os,s, Ast sstm st u sstm d uţ ortogo ş : s d os d ~ T 95
2 Fd dtă o uţ ), : R R, prodă u prod, s r să s dtrm odţ p r tru să îdpsă uţ prodă ) st îât să putm ostru sr trgoomtră ), uorm ovrgtă p,, d ş p R Î st potz putm sr gtt : [ ] 4) ) + os + s ) Sr d uorm ovrgtă, putm tgr trm u trm ş î z ortogotăţ sstmuu 3) găsm : 5) o ) d Îmuţd sr 4) u os ş tgrâd, oţm : )os d os d, d ud: 6) ) os Prodâd og, pr îmuţr u d s, oţm : 7) ) s d Coţ,, {,,3,} dtrmţ după ormu 6) ş 7) s ums oţ Fourr ptru uţ ) r sr trgoomtră ) u şt oţ s umşt sr Fourr uţ prod ) Fd dtă o uţ prodă u prod ş tgră, putm dtrm oţ Fourr orspuzător uţ dt prum ş sr Fourr tştă u ) Nu putm îsă să srm gtt 4) dor u ştm dă sr st ovrgtă ş hr î z d ovrgţă, u ştm dă sum st tom uţ D st motv vom sr : 8) ) + os + s ) Codţ sut ptru o uţ prodă u prod să potă rprzttă pr sr Fourr sotă, u ost găst d Drht Ar o: Torm Codţ u Drht) Dă uţ ) u prod st mootoă p porţu ş mărgtă p trvu[, ], tu sr Fourr sotă st uţ st ovrgtă î tot put Sum S) sr Fourr î r put d otutt st gă u vor uţ î put Î put d dsotutt, vor sum S) st gă u md rtmtă mtor tr orspuzător putuu d dsotutt, dă: 96
3 9) ) + + ) ) ) m ), + ) m ) S ud, < > Empu Cosdrăm uţ ), [, ] 4, Fuţ prodă grtă d uţ ) v trsormt prodă u prod ăr gr st : y 3 3 Fuţ ) rprztă rstrţ uţ ~ trvu [, ] Codţ torm u Drht sut îdpt, dor uţ p trvu [, ] st mootoă ş st mărgtă Apâd d două or tgrr pr părţ oţm ptru oţ Fourr prs : st : ),,, D sr Fourr orspuzător uţ Cosdrâd + 4 ) oţm sum: 6 os os os + + ) î trvu [, ]
4 Sr Fourr uţor pr su mpr Dă uţ ) st pră su mpră p[, ] tu dzvotr î sr Fourr s smpă Ast, dă uţ ) st pră p[, ], tu -) ) ş î osţă uţ ) os st pră r uţ ) s st mpră Ţâd sm d st vom oţ: ) )s d, ) os d ) d ) os d ) d Ptru uţ pr p [, ] sr Fourr v oţ um trm î osusur, dă trm pr D sr Fourr v v prs: os, + ) ) vă î put d otutt uţ ) p, ) Ast z ost ustrt pr mpuu d prgru tror pră p[, ] Oy ă d smtr) Dă uţ ) st mpră p trvu [, ] ), r st o uţ 4, tu uţ ) os st mpră, r ) s st o uţ pră Î osţă oţ sr Fourr vor : 3) o, ş ) s d Sr Fourr ptru uţ mpr v oţ um trm î susur, d : 4) ) s 3 Dzvotr î sr Fourr uţor dt p -, ) Empu Vom osdr zu gr dzvotăr î sr Fourr u uţ prod u prod T >) Şru trgoomtr udmt, v : ),os,s,,os,s, 98
5 F ) rstrţ uţ prod u prod T p trvu -, ) t t Etuâd shmr d vră, uţ v o uţ prodă u prod Rstrţ trvu, ) v uţ Srd dzvotr î sr uţ t t, vm : ) + os t + s t), vă î or put d otutt t R Dtortă susttuţ t / oţ Fourr vor v prs: 3) t dt )os d )s d ) d ) d D sr Fourr ptru uţ ) p trvu,) v : 4) ) + os + s ) t, ud oţ sut dţ d ormu 3) Empu Să srm sr Fourr orspuzător uţ ) p trvu -, ) Fuţ st mpră p -, ) d sr Fourr v oţ um trm î sus Avm : +, s d s d ) Pr urmr, sr Fourr orspuzător uţ ) v : Ptru + ) s , oţm sum : 4 Dzvotr î sr Fourr după osusur su susur u uţ dt p trvu, ) Empu F ) o uţ dtă p [ ], Dsor st ut uţ ) să s dzvot î sr Fourr după osusur su susur Î st sop uţ s 99
6 prugşt p trvu [,] st îât ou uţ F) să uţ pră su mpră p trvu [, ], după um dzvotr î sr Fourr tru să după osusur su susur Să prsupum ă dorm să dzvotăm uţ ), î sr Fourr după osusur gur): y -) ) - - Etuăm prugr pră p trvu [,], d uăm smtru gruu uţ î rport u ordotor Oţm st o ouă uţ F) pră p[,] ), [,] F ) ), [, ] Dă uţ dtă ) îdpşt odţ u Drht p trvu [, ], tu ou uţ F) v îdp st odţ p trvu [-, ] Pr urmr, sr Fourr orspuzător uţ F) v : ) F ) + os ud ) / F ) d ) d F )os d ) os d, Dzvotr ) r o î tot put d otutt d p trvu -, ) Î prtur, p trvu, ) oţm dzvotr ăuttă după osusur :
7 3) ) + os, vă î put d otutt d trvu, ) Aog ptru oţ dzvotr î sr Fourr după susur uţ ) dtă p [, ), tuăm o prugr mpră uţ p trvu [-, ) gur) : y - - ) --) ş oţm st o ouă uţ : ), [,] F ) ), [, ] Astă uţ st mpră p trvu [-, ],gru d smtr î rport u org sstmuu d rrţă Srd dzvotr î sr Fourr ptru uţ mpră, vom oţ : 4) F) s ud: 5), F )s d, su )s d
8 Î prtur, î or put d otutt d trvu, ) vm dzvotr după susur uţ dt ), um: 6) ) s Empu Să dzvotăm î sr Fourr după susur uţ )-, [, ) Etuâd o prugr mpră p trvu -, ) ) uţ dt, vom oţ uţ:, [,) F ), [,] Pr prodzr uţ F) s oţ gru : y ud : Î osţă, sr Fourr uţ osdrt v - D: )s s - s 5 Form ompă sror Fourr O ormă utră sror Fourr st orm ompă F ) o uţ r p trvu -, ) sts odţ torm u Drht Atu putm sr dzvotr î sr Fourr :
9 3 ) ) s os ) + +, ud oţ sr u prs: ) d d d )s )os, ) Utzâd ormu u Eur: 3) ) ),s os +, sr ) dv: 4) ) ) + + Ţâd sm d prs ) oţor vm : 5) d ) ş 6) - d ) Rmrăm ă î 5) ş 6), N * Prmu trm dzvotăr ) r prs : 7) ) d r s oţ d 5) ptru Pr urmr, sr 4) s pot sr su orm : 8) ) + su 9) ) ud ) d ), Z Eprs 9) d rprztr uţ ) s umşt orm ompă sr Fourr 6 Dzvotr u uţ î sr d uţ ortogo Apromr uţor î md pătrtă Rţ d îhdr u Prsv Azîd modu d dtrmr oţor sr Fourr, osrvăm ă rţomt oost u s-u zt p proprtăţ ort uţor
10 trgoomtr d sstmu trgoomtr udmt um p proprtt d ortogott D st motv st tur î ou sstmuu trgoomtr d uţ ortogo să uăm u sstm orr d uţ ortogo Î st o uţ pot rprzttă î sr u u sstm d uţ ortogo, oţâd o sr Fourr grztă F şru d uţ ortogo ϕ )) L, ) d pătrt tgr p,) R ) Ptru smpr uor vom prsupu ă şru ost ormzt ş vom ot u Ψ )) şru ortoormt d L,) Să prsupum ă L,) ş ă s pot rprzt su orm u sr uorm ovrgt p,) î rport u sstmu d uţ ortoormt Ψ )) Coorm potzor ăut vm : ) ) ) Ψ Ptru dtrmr oţor N), îmuţm gtt ) u ojugtu Ψ uţ Ψ ş tgrâd trm u trm p trvu,), oţm : ) ) Ψ d Ψ Ψ d Ψ ş dor sstmu Ψ ) st ortoormt, vm : 3), m Ψm, Ψm ), m Coţ dtrmţ pr rţ ) s ums oţ Fourr grzţ uţ L,) rtv sstmu ortoormt d uţ Ψ ) p, ) Sr ) s v um sr Fourr grztă uţ rtv sstmu ortoormt Ψ ) Torm u Drht rămâ vă ş ptru sr Fourr grzt Ast rţ ) r o î r put d otutt uţ d trvu, ) dă prt ră ş prt mgră uţ omp L,) sts odţ torm u Drht Empu Să dzvotăm î sr după poom u Hrmt uţ ), R Poom u Hrmt dt pr rţ: 4) ) ortogo u podr p) Fuţ ) : d H ) ) N R L R) d p R 5) H ), R, ormză u sstm ş sts odţ torm u Drht, d 4
11 Îmuţd stă gtt u H ) ş tgrâd, p z proprtăţ d ortogott oţm : + H )! st : + d H ) d H ) d! d ud: Itgrîd pr părţ ş ţâd sm d 4) oţm: + + H ) d H ) d + 4 d Pr urmr, sr Fourr grztă orspuzător uţ ) 4 H ),! R L, vă ptru or Dţ F,g ) Numm ror pătrtă md uţ ţă d g umăru 6) δ ) g ) d ) g ) Număru δ rprztă o măsură ror o m dă promăm uţ pr g su uţ g pr Astă măsură ror umtă ror pătrtă md st dost d ută î studu sror Fourr, dor st gtă drt d orm uţor d pătrt tgr F uţ L, ) ş sstmu ortoormt d uţ omp Ψ d pătrt tgr p trvu,) Fuţ: )) 7) S ) λ Ψ ) s umşt poom ortogo p trvu, ) Să dtrmăm oţ λ poomuu 7) st îât ror pătrtă md ţă d uţ să mmă Avm : δ ) ) S ) d ) λ Ψ d Ţîd sm ă uţ, Ψ sut uţ omp, r λ umr omp, ptru dzvotr prs d su smu tgră d m sus vom oos ormu : Oţm : α β α β ) α β ) α + β α β αβ 5
12 8) δ ) d λ Ψ d λ Ψ d + λ λ Ψ Ψ d j j j Sstmu d uţ Ψ d ortoormt ş ţâd sm ă oţ Fourr orspuzător uţ rtv sstmu ortoormt Ψ ) sut ) Ψ d gtt 8) dv: 9) δ ) λ λ + λ λ + + λ ) λ ) λ λ D rţ 9) rzută ă δ v mmă dă Am oţut st : Torm Dtr tot poom ortogo, ptru r ror pătrtă md ţă d uţ L, ) st mmă st ăru oţ sut oţ Fourr grzţ rtv uţ Ast îsmă ă uţ Ψ rzză m uă promţ î md pătrtă uţ d pătrt tgr Putm sr: ud ) δ ) Dor δ, rzută gtt: ) d ), umtă gtt u Bss Putm st uţ : Torm Sum pătrtor moduor oţ Fourr u uţ d pătrt tgr, rtv u sstm d uţ ortoormt, st mut gă u pătrtu orm uţ Dă osdrăm sr u trm poztv tu d gtt u Bss ddum ă sum prţ sr sut mărgt d ; pr urmr sr st o sr ovrgtă D st motv î gtt u Bss putm osdr ş s oţ : ) umtă gtt u Prsv + 6
13 Dţ U şr ortogo d uţ Ψ) d pătrt tgr st u sstm îhs, dă ptru or L, ) r o rţ : 3) umtă rţ d îhdr u Prsv F L, ) > Sstmu trgoomtr ormt : os s os s 4),,,,,, st u sstm îhs Î rport u sstmu ortogo 4) oţ Fourr sut : / os ) d ) os d // ) s d ) / // Îoud,, oţuţ m sus î 3), oţm rţ d îhdr u Prsv ) d 5) + + ) Dă, 5) dv : ) d 6) + + ) Empu Să s sr sr Fourr trgoomtră ş po gtt u Prsv ptru uţ:, ptru < ), ptru < Să s dduă po sum sror: Sr Fourr st: ) ) + ud os + s s ) ş ) ) d, )os d, ş os ) s d, 7
14 Gru u ) st: y - - Avm 3) Apo, 4) d, d ud rzută: d s os s s,dă: ş: s d os dă: 5) ) pră!) D sr Fourr tştă uţ z) st: 6) ) + Egtt u Prsv st: s os su 7) + + ) ) d d ud: 4 8) + s d + s 9) Rzută sum rută: 8
15 ) s Ptru u os srm: os s s Ştm ă: 6 ;d os 6 7 Prom propus ) Să s dzvot î sr Fourr uţ : ) ),,] ) ; 3,, ], [,] ),,) ; 3, [,3] os 5 + 4os ) ), R ) Să s dzvot î sr Fourr d s ş rsptv os uţ : ) ),, ) ; 4 ), [,] ),,] 9
16 3) Să s dtrm sr Fourr trgoomtră uţ prod ),, ) sh d prodă D dzvotr oţută ş d rţ d îhdr u Prsv să s uz sum : ) + ş + 4) Să s sr sr Fourr trgoomtră ş po gtt u Prsv ptru uţ :, ), <, > Să s uz po sum sror : s ş os
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