A Study on the Bases of Space of Vector Valued Entire Multiple Dirichlet Series
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1 A Study o te Bses of Spce of Vecto Vued tie Mutipe Diicet Seies Mustq Si A Hussei d Nge R Nge * Deptet of Mtetics Coege of Sciece Uivesity of Mustsiy Bgdd Iq Abstct: Let f ( s s e ( s s ( s it beig icesig sequeces of positive ubes d d wee is Bc geb epeset vecto vued etie Diicet fuctios i two vibes Te spce of suc etie fuctios vig ode t ost equ to is cosideed i tis ppe A etic topoogy usig te gowt petes of f is defied o d its vious popeties e obtied Te fo of ie opeto o te spce is ccteized d pope bses e so ccteized i tes of gowt petes Keywod: Vecto Vued Diicet Seies Bc Ageb tie Fuctio د ارسة عن اساسات فضاء متجه القيم لسلسلة درشلت الكلية المتعددة مشتاق شاكر و نجم رعد نجم قسم الرياضيات كلية العلوم الجامعة المستنصرية بغداد الع ارق ( s it f ( s s e ( s s الخالصة ليييييييك متتاليييييات مت اةيييييدق ميييي ا رقييييام الموجبيييي و حييييي و هييييو بنييييا الجبيييي ار يمثييييا دواا درسينا ييه هياا البحي درشلت الكليية متجةية الميمي لمتغييري لمييد ن رينييا ال ضييا المتيير باسييتخدام نمييو متغييي ارت الداليية يضيا جمييا اليدواا الكليية اوات الرتبي والمع ن رييية لييف ال ضييا وح ن صييلنا لييف f شكا المؤثر الخطه لف ال ضا والممثا باألساس ال عله بد لة نمو الوسيط *i: ged@yooco 749
2 Itoductio Let f ( s s e ( s s s it wee ( t e e vibes ( Wee s beog to couttive Bc geb wit idetity eeet wit d s R stisfy te coditios s so s R stisfy te coditios s d og i sup ( i sup D (3 Te te vecto vued Diicet seies i ( epesets etie fuctio f ( s s (see [] d [] I [] GS Sivstv d Ac S defied te gowt petes suc s ode type of vecto vued Diicet seies i two vibes Tey so obtied te esuts fo coefficiet ccteiztio of ode d type Te cocepts of ode d type of etie fuctio (so fo ytic fuctio epeseted by vecto vued Diicet seies of oe copex vibe wee fist itoduced i 983 by BL Sivstv [3] Tey so obtied te coefficiet ccteiztios of ode d type Te spce Y of etie fuctios epeseted by vecto vued Diicet seies f (s of oe copex vibe vig ode t ost equ to wee fist itoduced i by GS Sivstv d Ac S [4] A etic topoogy usig te gowt petes of f is defied o Y d its vious popeties e so obtied by te Tey so obtied te fo of ie opeto o te spce Y is ccteiztio d pope bses e so ccteiztio i tes of gowt petes I tis ppe we ve exteded d ipove te bove esuts to te etie fuctio epeseted by vecto vued Diicet seies of seve copex vibes Fo te se of sipicity we coside te fuctios of two copex vibes Toug ou esuts c be esiy exteded to fuctios of seve copex vibes Let fo etie fuctios defied s bove by ( M ( sup f ( it it t t Te M ( is ced xiu oduus of f ( s s o te tube Re s Ji d Gupt [5] defied te ode ( of f s s s: ( og og M ( i sup e e ( Let us deote by te ie spce of vecto vued etie utipe Diicet fuctios f of fiite ode ess t o equ to Te evey fuctio f stisfyig i sup D (4 is ccteized by te coditio i sup og Tis is equivet to te coditio /( (5 / / /( ( s fo ec (6 Now fo ec f fo we defie te qutity /( ( f (7 I view of (6 f is we defied d defies o o Let us deote by ( Te spce equipped wit te o f Tis o iduces etic topoogy o We c defie te equivet etic 75
3 d( f g f g; ( q t qt q t f g; ( q t f g (8 Toug tis ppe we s ssue tt is equipped wit te topoogy geeted by te etic d Foowig Suze D d Poities [6] we give soe defiitios A Sequece of fuctio is sid to be iey idepedet if c ipies tt c fo sequece of copex ubec fo wic c coveges i A subspce of is sid to be sped by sequece if cosists of ie cobitios c suc tt c coveges i A sequece wic is iey idepedet d sps subspce of is sid to be bse i I pticu if e e ( s s s s e e te is bse i A sequece wi be ced pope bse if it is bses d it stisfies fo sequeces covegece of i ipies te covegece of e i Fist we s pove Teoe Te spce is copete wit espect to te etic d Poof: Letf be Cucy sequece i wee f ( s s e ( s s Hece it is Cucy sequece i ( Teefoe fo give positive ube d tee exists positive itege N N suc tt ( ( ( f f N /( ( ( f f Deotig by f ( s s e ( s s f s s ( e s s ( We ve teefoe ( ( ( /( N ( Teefoe fo ec fixed 3 ( is Cucy sequece i te Bc spce Hece tee exists sequece suc tt ( i Now ettig N i ( we ve fo ( /( ( ( Now we coose Te we ve ( N ( N Hece by (6 we ve /( ( /( /( /( /( ( /( ( ( /( ( Sice bity d teefoe we obti i /( ( fo ec Tus f ( s s e Teefoe f f i Hece is copete Hee 75
4 is oed ie etic spce d is copete wit espect to te etic d d ece it is Fecet spce so Tis poves Teoe Next we pove Teoe A cotiuous ie fuctio : ( is of te fo if d oy if ( f c f e c is /( ( bouded fo Poof: Let ie fuctio o ( be give by ( ( f c f e wee c e Let be cotiuous Hece tee exists positive costt suc tt ( f f fo f s s Assuig f e e tis ipies tt /( c ( Covesey et f be s bove d ( f c wee c is bouded Hee ( f /( ( is we defie sice c ( c /( Teefoe is cotiuous ie fuctio o ( We ext pove Teoe 3 Let (i d be s bove Te te foowig e equivet: Tee exists cotiuous ie tsfotio T : wit T ( e 3 3 (ii Fo ec og ; i sup (3 Poof: Let T be cotiuous ie tsfotio fo ito wit ( 3 3 Te T e fo y give tee exists d costts K K ( K K ( depedig o d espectivey suc tt o Hece T ( e ; e ; ( ; /( /( og ; i sup Covesey et te sequece stisfy (3 d et wit ( s s e Te it foows tt i O /( / ( / ( fo ec /( / ( / ( fo N Fute fo give we c fid N N( N N( fo (3 suc tt fo og ; ( fo N N O 75
5 /( /( ; fo N N Coose x( N N x( N N Te /( /( /( ; ( /( ( Sice te seies /( ( /( ( ; coveges fo ec Sice coveges bsoutey i d sice is copete we fid tt coveges to eeet of Hece tee exists tsfotio T : suc tt T( (sy fo ec We obseve tt T is ie d ( Now we ve oy to T e pove te cotiuity of T Fo (3 give tee exists suc tt og ; fo ( N N( ( ; /( /( ( ( Hece T( ; ; /( /( fo ; ( Hece T is cotiuous Tis poves Teoe 3 We ow give te ccteiztio of pope bses Teoe 4: Let d be give sequeces Te foowig tee coditios e equivet: (i (ii Covegece of e i ipies te covegece of i Te covegece of e i ipies tt i ( i (iii og ; i sup fo ec Poof: Fist suppose tt (i od Te fo y sequece wee s beog to Bc geb e coveges i ipies tt coveges i wic i tu ipies tt s Hece (i (ii Now we ssue tt (ii is tue but (iii is fse Tis ipies tt fo soe tee exists sequeces of positive iteges suc tt og ; ( d Defie sequece s 753
6 /( /( (4 Te we ve ( /( /( ( /( ( /( ( fo sufficiety ge d wit Hece i sup /( /( d teefoe e coveges by (6 But ; does ot ted to zeo s wic cotdicts (ii Hece (ii(iii I couse of te poof of Teoe bove we ve edy poved tt (iii(i Tus te poof of Teoe 4 is copete Teoe 5: Let d Te foowig tee popeties e equivet: ( i ( i ipies tt e coveges i (b Covegece of i ipies tt e coveges i (c i i og ; if Poof: It is evidet tt ( (bwe ow pove tt (b (cto pove tis we suppose tt (b od but (c does ot od Hece og ; i i if Sice ; iceses s deceses it foows tt fo ec og ; i if Hece if be fixed s positive ube te fo ec we c fid positive ubes suc tt we ve og ; ( d ( (5 Now we coose positive ube d d defie sequece s /( /( (6 Te fo y ; ; Fo y give oit fo te bove seies tose fiite ube of tes wic coespod to tose ubes fo wic / d / Te eide of seies i (6 is doited by ; ( Now by (5 d (6 we fid tt ; ( /( /( /( /( ( /( ( ( /( ( 754
7 Sice ece te bove seies ; ( is coveget Fo tis sequece coveges i ( fo ec d ece coveges i But we ve /( /( /( /( /( /( /( /( ( /( ( ( /( ( Now we coose d te fo bove /( does ot /( ted to zeo fo tis Hece e does ot covege d tis is cotdictio Teefoe (b(c Now we pove tt (c( We ssue (c is tue but ( is ot tue Te tee exists sequeces fo wic i but e does ot covege i Tis ipies tt i sup og Hece tee exists positive ube d sequece of positive iteges suc tt ( og (7 We coose ote positive ube / / by ssuptio we c fid positive ube ie ( suc tt og ; i if ( Hece tee exists N N( suc tt og ; ( (8 Teefoe fo (7 d (8 we ve fo ; ; /( /( ( /( ( /( /( ( /( ( s Sice Te does ot ted to zeo i ( fo te cose bove Hece does ot ted to zeo i d tis cotdictio Tus (c(tis poves Teoe 5 Cooy A bse i cosed subspce of is pope bse if d oy if it stisfies te coditio (iii d (c of Teoe 4 d Teoe 5 espectivey Refeeces SuzeD tie fuctios epeseted by Diicet seies of two copex vibes Potugie Mt Vo43 Fsc4 pp: Sivstv G S d Ac S Soe gowt popeties of etie fuctios epeseted by vecto vued Diicet Seies i two vibes Ge Mt Notes Vo No pp: Sivstv B L 983 A study of spces of ceti csses of vecto vued tie Diicet seies Tesis Idi Istitute of Tecoogy Kpu 4Sivstv G S d Ac S Bses i te spce of vecto vued tie Diicet seies vig fiite ode It J Cotep Mt Scieces Vo 7 o pp:
8 5Ji PK d GuptVP 974 O ode d type of etie Diicet seies of seve copex vibes pp: Suze D d Poities 984 Bses i te spce of etie Diicet fuctios of two copex vibes Coect Mt 35 pp:
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