TEOREMA ABEL-DINI DAN DUAL KÖTHE-TOEPLITZ PADA DERET GANDA

Ukuran: px

Mulai penontonan dengan halaman:

Download "TEOREMA ABEL-DINI DAN DUAL KÖTHE-TOEPLITZ PADA DERET GANDA"

Bambang Tedja
6 tahun lalu
Tontonan:

Prosdg Semr Nsol Ss d Peddk Ss VIII, Fkults Ss d Mtemtk, UKSW Sltg, 5 Ju 203, Vol 4, No, ISSN:2087 0922 TEOREM BEL-DINI DN DUL KÖTHE-TOEPLITZ PD DERET GND Sumrdoo, Soer DW 2 & Sum 3 PPPPTK Mtemtk,

Köthe-Toeltz, Deret Gd PENDHULUN Pd stud lss, dkel beber teorem g m dktk deg m bel muu D Slh stu teorem g berkt deg deret tk hgg dkel deg m Teorem bel-d Beber ltertur g membhs tetg Teorem bel-d tr l

1 Prosdg Semr Nsol Ss d Peddk Ss VIII, Fkults Ss d Mtemtk, UKSW Sltg, 5 Ju 203, Vol 4, No, ISSN: TEOREM BEL-DINI DN DUL KÖTHE-TOEPLITZ PD DERET GND Sumrdoo, Soer DW 2 & Sum 3 PPPPTK Mtemtk, Mhssw S3 UGM smrd2007@gmlcom 2,3 Jurus Mtemtk, FMIP, UGM 3 sum@ugmcd BSTRK Pd er Teorem bel-d derlus d deret gd Drk ul lks d Dul Köthe-Toeltz sutu Rug Brs Gd Kt-kt kuc: Teorem bel-d, Dul Köthe-Toeltz, Deret Gd PENDHULUN Pd stud lss, dkel beber teorem g m dktk deg m bel muu D Slh stu teorem g berkt deg deret tk hgg dkel deg m Teorem bel-d Beber ltertur g membhs tetg Teorem bel-d tr l Hldebrdt [5], Rjgol [9], d Ko [7] Semetr kose dul Köthe-Toeltz telh drts oleh Köthe & Toeltz [8], Grlg [3],Chllgworth [2], d msh bk lg Tok jug dbhs deg cuku legk dlm Kmth & Gut [6] d Wlsk [0] Kose dul berbed deg dul toologs g ggot meruk fugsol kotu Pd er, ots tu metk jumlh dr = sm deg tk hgg Jk terbts mk bts ts k dtk d ots sgm Berkut Teorem bel-d utuk deret tuggl blg rel Teorem ([], [7] Jk dlh deret rel o-egtf g dverge d = = mk koverge utuk > 0 (d dverge utuk 0 Pegert dul-α dtk sebg berkut Defs Dberk rug vektor tk ol λ ω deg ω dlh koleks semu brs blg rel (tu komlek mk ddefsk dul-α sebg berkut λ α = { : ω,, λ} Sel dul-α, jug dkel dul-β, dulγ, d dul- dr λ Nmu dlm er h k dkj g berkt deg dul-α Kesemu dul rug vektor d ts, terutm dul-α, serg dsebut deg dul Köthe-Toeltz Seljut k dbhs erlus Teorem bel-d d rug brs gd, d ter dlm meetuk dul Köthe-Toeltz d sutu rug brs gd 457

2 Prosdg Semr Nsol Ss d Peddk Ss VIII, Fkults Ss d Mtemtk, UKSW Sltg, 5 Ju 203, Vol 4, No, ISSN: TEOREM BEL-DINI PD DERET GND Pd stud, eleme brs dbts d blg rel Perlus ke eleme blg komleks mudh dkostruks Brs gd (double seuece meruk sutu geerlss dr brs tuggl (sgle seuece Defs 2 Brs gd ddefsk sebg fugs sebg berkut : N N R Nots ( metk brs gd deg R,, j Seljut utuk membuktk Teorem bel-d d deret gd, derluk beber lemm sebg berkut Lemm ([], [4] Jk, r R, > 0, mk r > r ( utuk r > r < r ( utuk 0 < r < Lemm 2 ([], [4] Jk, R,, r R, mk ( r r ( > r r > r r ( utuk r < 0 tu r > (b r r ( < r r < r r ( utuk 0 < r < Sekrg, kt s utuk memerlus Teorem bel-d d deret gd Teorem 2 (Teorem bel-d utuk Deret Gd deret dverge rel Jk j m o-egtf d m = mk > 0 j = j = koverge utuk Utuk 0 < < Ddg deret j ( ( ( ( = ( j ( ( j Jumlh ( d, kre bl, j mk Meurut Lemm 2 mk > ( ( = Sehgg deroleh j ( ( < ( ( ( j ( ( ( ( Kre rus k dlh deret ( g koverge d berdsrk uj bdg, mk deroleh koverge j Lbh ljut, ( ( j ( ( = j ( ( = j ( ( Kre > ( (j mk j < j ( ( Sehgg deroleh Bukt: 458

3 Prosdg Semr Nsol Ss d Peddk Ss VIII, Fkults Ss d Mtemtk, UKSW Sltg, 5 Ju 203, Vol 4, No, ISSN: j j Kre 0 mk j Utuk > Meurut Lemm 2 mk ( < tu (koverge ( ( ( ( ( ( Sehgg deroleh < ( ( ( ( ( ( ( ( ( ( ( ( < ( ( < ( ( Kedu rus dkek double sgm sehgg deroleh j ( ( < j ( ( j Rus k koverge, kre utuk, j mk Deg Uj Bdg deroleh j ( ( Oleh kre > ( (j mk deroleh j < Sehgg deroleh j ( ( j D kre 0 mk deroleh j (koverge DUL KÖTHE-TOEPLITZ PD RUNG BRISN GND Pd bg, k dbuktk dul-α sutu rug brs gd deg megguk Teorem bel-d Nots 2 ω d 2 l berturut-turut metk rug semu brs gd, d brs gd g terjumlh mutlk (-bsolutel summble dr blg rel 2 l = {( 2ω : } deg Defs 3 Dmslk E hmu bg tk kosog dr 2 ω, mk dul-α dr E ddefsk deg E α = {( 2ω : b j utuk set( E} j Utuk meetuk dul-α beber rug brs gd, derluk lemm d teorem berkut Lemm 3 (Ketksm Hölder utuk 2 l Utuk set,( 2 l mk berlku ( Bukt: ( j j ( j 459

4 Prosdg Semr Nsol Ss d Peddk Ss VIII, Fkults Ss d Mtemtk, UKSW Sltg, 5 Ju 203, Vol 4, No, ISSN: Dbetuk α = = ( j ( j d β deg = Mk meurut Lemm Youg, deroleh ( j ( j ( j ( j Kedu rus dmbl double sgm, deroleh ( ( j j ( j j j j ( j ( ( j j j j ( ( j (terbukt Teorem 3 j Jk j m o-egtf d m = mk Bukt: j deret dverge rel = j = dverge Utuk set m,, r, s Z (hmu semu blg bult ostf, mk (m m sehgg deroleh ( m ( ( m ( 2 ( m ( ( 2( 2( ( 2 2( 2 2( 2 ( 2( 2( 2 ( m ( 2( 2 ( m ( 2 ( m 2( 2 ( 2( 2 ( m m = m = F tu dtetk m d sehgg m utuk r, s mk I rt = m j= set m, N sehgg lm, j > utuk 0 Deg demk deroleh bhw deret tdk koverge tu j dverge Terkhr, dbuktk dul-α rug brs gd 2 l megguk Teorem bel- D d Deret Gd Teorem 4 ( l α 2 = 2l deg > d = Bukt: 460

5 Prosdg Semr Nsol Ss d Peddk Ss VIII, Fkults Ss d Mtemtk, UKSW Sltg, 5 Ju 203, Vol 4, No, ISSN: Dmbl sebrg ( 2 l mk j Utuk set ( 2 l deg = mk deroleh j ( j ( j l α 2 2 l Seblk, Jd, ( Dmbl sebrg ( α ( l 2 Ddk ( 2 l mk j = Dmslk = d Dbetuk j m m = = j = mk = = (dverge d b = Dbetuk ( deg = mk deroleh = b = Kre 0 (o-egtf d = mk j ( Meurut Teorem 2 deroleh j = j = j Jd, { ( } l (ω ( Meurut Teorem 3 deroleh j = b j j b = b = j = (kotrdks Sehgg hruslh ( 2 l l α 2 l Jd, ( 2 KESIMPULN Teorem bel-d dt derlus deret gd (double sere Teorem jug meruk slh stu lt g dt derguk utuk meujukk dul-α sutu rug brs gd, khusus rug brs 2 l DFTR PUSTK [] Bccl, Bret 2008 The Hölder d Mkowsk Ieultes lecture er, ot ublshed [2] Chllgworth, HR 958 Geerlzed dul seuece sces Nederl kd Wetesch Idg Mth 20 [3] Grlg, DJH 967 The β- d γ- dult Proc Cmbrdge Phlos Soc 63 [4] Hrd, G, Lttewood, JE, & Pól, G 934 Ieultes Cmbrdge: t The Uverst Press [5] Hldebrdt, TH 942 Remrks o the bel-d Theorem The merc Mthemtcl Mothl Vol49, No7 (gt-se, 942, hl [6] Kmth, PK & Gut, M 98 Seuece sces d seres New York Mrcel Dekker, Ic [7] Ko, Kord 956 Ifte Seueces d Seres Trslted b F Bgemhl New York: Dover Publctos, Ic [8] Köthe, G & Toeltz 934 Lere Räume mt uedlch vele Koordte ud Rge uedlcher Mtrze, Jour Ree gew Mth 7 [9] Rjgol, CT 944 The bel- D d lled Theorems The 46

6 Prosdg Semr Nsol Ss d Peddk Ss VIII, Fkults Ss d Mtemtk, UKSW Sltg, 5 Ju 203, Vol 4, No, ISSN: merc Mthemtcl Mothl Vol5, No0 (Des, 944, hl [0] Wlsk, 984 Summblt through Fuctol lss msterdm: North Holld 462

7 Prosdg Semr Nsol Ss d Peddk Ss VIII, Fkults Ss d Mtemtk, UKSW Sltg, 5 Ju 203, Vol 4, No, ISSN: Nm Pe Ists Pert : : H P : UKSW kh reesets dr deret tersebut dlh mtrks d kergm k koverges? Jwb : Y, d keverges beru sklr Nm Pe : M Mhfuzh S Ists : Uv Lmbug Mgkurt Pert : kh rug sel deret gd bs megguk teorem bel-d? Jwb : Tdk bs, kre bel D megguk kose brs Pert : Volume tu ukur utuk? Jwb : - Nm Pe : Trev Mer dr Ists : UKSW Pert : Trf Sgfks seert? Jwb : Bts elug 463

dokumen-dokumen yang mirip

BAB 1 PENDAHULUAN. 1.1 Latar Belakang

BAB 1 PENDAHULUAN. 1.1 Latar Belakang BAB PENDAHULUAN. Ltr Belkg Smp st, model Regres d model Alss Vrs telh dpdg sebg du hl g tdk berkt. Meskpu merupk pedekt g umum dlm meergk kedu cr pd trf permul, model Alss Vrs dpt dpdg sebg hl khusus model