Conversely Convergence Theorem of Fabry Gap

Transcription

1 Sciece Joural of pplied Matheatics ad Statistics 5; 3(4: Published olie Jue 3 5 ( doi: 648/jsjas534 ISSN: (Prit; ISSN: (Olie Coversely Covergece Theore of Fabry Gap Naser bbasi Molood Gorji * Departet of Matheatics Faculty of sciece Loresta Uiversity Khoraabad Islaic Republic of Ira Eail address: aserbbasi_persia@yahooco (N bbasi olodgorji@yahooco (M Gorji To cite this article: Naser bbasi Molood Gorji Coversely Covergece Theore of Fabry Gap Sciece Joural of pplied Matheatics ad Statistics Vol 3 No 4 5 pp doi: 648/jsjas534 bstract: Our previous paper coducted to prove a variatio of the coverse of Fabry Gap theore cocerig the locatio of sigularities of Taylor-Dirichlet series o the boudary of covergece I the preset paper we prove coversely covergece theore of Fabry Gap This is aother proof of Fabry Gap theore This prove ay be of iterest i itself Keywords: Dirichlet Series Etire Fuctios Fabry Gap Theore Itroductio The Fabry Gap theore ([] states that if Λ { is a real positive sequece such that c for c > ad D as the the Dirichlet series f ( z c e z has at least oe sigularity i every iterval of legth exceedig π D o the abscissa of covergece We assue that the reader is failiar with the theory of Etire Fuctios ad the theory of Dirichlet series as used i the books [69 ] We ote that other results cocerig the locatio of sigularities of Taylor Dirichlet series have bee derived by Blabert Parvatha ad Berlad (see [3 5] uxiliary Results ad Notios I this sectio we describe the defiitios ad also to express ad prove the lea we eed to prove the theore Defiitio We deote by L ( c D the class of all sequeces { a with distict coplex ters a divergig to ifiity a a satisfyig the followig coditios: (see also [] ( There is a costat c > so that a a c for all ( li / a D (3 the sup arg a < π / Defiitio Let the sequece L ( c D ad real positive ubers so that < We say that a sequece B { b with real positive ters b ot ecessarily i a icreasig order belogs to the class if for all N we have b { z C : z a a ad for all oe of the followig holds: ( i b b a a ( ii b b ax{ e e Oe observes that ( i allows for the sequece B to have coicidig ters lso ote that ( ii allows for o-coicidig ters to coe very close to each other We ay ow rewrite B i the for of a ultiplicity sequece Λ { by groupig together all those ters that have the sae odulus ad orderig the so that < We shall call this for of B the ( reorderig (see also [] give the sequeces L ( c D ad B (see also [4]: a e S B ± a ( 3

2 78 Naser bbasi ad Molood Gorji: Coversely Covergece Theore of Fabry Gap where as usual S a e B ± b ( 3 B( z r { z : z z < r Observe that the disks i S is ot ecessarily disjoit sice for fixed we ight have b b for We state ow leas that were proved i [] regardig ultiplicity sequeces They shall be used later o Lea 3 Let L( c D be a real positive sequece ad let B so that B { b is real positive too with ( its reorderig The the regios of covergece of the three series where ad * ** f f f as defied i z f ( z p ( z e (3 j j p ( z c z is a polyoial with c j (4 * z ** z ( ( f z e f z z e are the sae For ay poit z iside the ope covex regio the three series coverge absolutely Siilarly if istead of a real sequece L( c D we have a coplex sequece L( c Lea 4 There exist positive costats ψ ad χ so that for ay oe has ( ψ a χ b Lea 5 For ay oe has χ { L Lea 6 Let be ayoe of the followig sequeces: { { / 3 { siγ or { η where η > so that η < The j j l ( j l! L l j j! c b b jl ( < Theore 7 (Phrage-Lidelof Let f ( z be aalytic i the regio betwee two straight lies akig a agle π / at the origi ad o the lies theselves Suppose that o the lies ad that as r f ( z M (5 r f ( z O ( e where < uiforly i the agle The (5 holds throughout the regio Theore 8 Let { a be a coplex sequece satisfyig L ( c D ad arga as Let B ad let ( be its reorderig The the etire fuctio z G( z (6 satisfies the followig for every ε > as r : iθ G( re O( exp{ πr( D siθ ε (7 i re θ S ad wheever O( exp{ πr( D siθ ε (8 i G( re θ Furtherore for every ε > as oe has:! G [ ] ( O( exp{ ε (9 Valiro [3] (p 9 proved that if B { is a ultiplicity-sequece that satisfies the relatios l li li ( the regios of covergece of the Taylor-Dirichlet series ad its two associate series f ( z p ( z e * z ** z ( ( f z e f z z e ( wher ax{ c : j are the sae j For ay poit z iside the ope covex regio the three series coverge absolutely 3 Mai Results Theore 3 Let L ( c D be a real positive sequece for D > Let B so that B { b is real positive too ad let ( be its reorderig The ay Taylor-Dirichlet series f ( z as i (3 satisfyig l c l li i ( sup l sup has at ost oe sigularity i every iterval of legth z

3 Sciece Joural of pplied Matheatics ad Statistics 5; 3(4: exceedig π D o the abscissa of covergece Proof We follow o the lies of the proof of Theore XXIX i [] Let f ( z ad f * ( z f ** ( z ad as defied i (3 ( ax{ c : j (3 j Fro Lea 3 the regios of covergece of the three series are the sae Sice the are real positive ubers we cosider the o-trivial case that is whe the three series coverge i idetical half-plaes of the for R z > x x R With o loss of geerality we assue that the abscissa of covergece (ordiary ad absolute is the lie x I other words the relatio L li sup (4 holds Thus all three series coverge absolutely ad uiforly i ay half-plae x τ > Oe also otes that fro ( we have L c (5 l isup Suppose ow that there exists a iterval of legth greater tha π D o the lie x o which f ( z has two sigularity x x The with o loss of geerality we ca also assue that this iterval is B y B where B > πd This iplies the existece of soe a a > such that $ f(z $ is aalytic for x a x a y B y B We put so that < γ γ < π / ad let a γ arcta (6 4πD a γ arcta (7 4πD b b ( B πd ta γ (8 ( B πd ta γ (9 The rest of the proof is broke ito three steps (to prove that x x ca be obtaied siilarly ll three steps ake use of the covergece of ** due to f ( b Step : Sice f ( z is regular i the sei-strip ( x a y B the for all w C so that R w < we defie a b ( ( ( a a H R w f z e dz f z e dz R f ( z e dz f ( z e dz b R R ( where R > b ad the paths of the itegrals are the segets joiig the various poits The we prove that H ( R w H w coverges as R ad if we deote this liit by ( oe has a b ( ( ( a a H w f z e dz f z e dz ( ( b j! ( j l! j l j l ( b ( w e cj l j l ( w ( But ow H ( w is well defied for all w C { I fact it is aalytic i C { Next we defie J ( w H ( w G ( w where G ( w is the etire eve fuctio defied i Theore 8 The J ( w is a etire fuctio i the coplex plae Proof step : Observe that the first two itegrals i ( are idepedet of R thus we deal with the other two We will prove that ( holds as R The absolute covergece of f ( z i the iterval [ b R ] justifies itegratig it ter by ter to get the followig: R b f ( z e dz ( ( R j! ( j l! j l jl ( R( w e cj l j l ( w ( ( b j! ( j l! j l jl ( b ( w e cj l j l ( w Deote by I( R w the ifiite series which depeds o R We will show that I( R w as R Sice R w < the w > It follows fro Lea 6 that ( ( R j! R l ( ( j l! j l jl c j j l w Hece ( b b e (

4 8 Naser bbasi ad Molood Gorji: Coversely Covergece Theore of Fabry Gap ( R( w I( R w R e e BIw R e R ( O the other had sice / oe has ( /4 ( R < e ad therefore BIw ((3 R/ 4 BIw ( R/4 R( / < e e e e I( R w e Fro relatio (4 oe gets that (3 li sup(l / ( / ad this iplies that the Dirichlet series *** ( / z ( e f z coverges absolutely for ay z if R z > Thus the series *** ( / R ( e f R is defied for all R > ad is a positive decreasig fuctio Therefore there exists soe M > so that for all R > oe has f *** ( R < M Cobiig this with (3 shows that I ( R w as R Siilarly oe deduces that R li f ( z e dz (4 R R Therefore for all w with R w < oe has that the li H ( R w exists If we deote this by H ( w the R H ( w takes the for as i ( Next we prove that H ( w is well defied for all w C { I fact we prove that H ( w is aalytic i C { Note that the two itegrals i ( defie aalytic fuctios of w i the whole coplex plae Thus it reais to be proved that the ifiite series coverges uiforly o ay copact subset K C such that K { Cosider such a copact K The there exists a N so that for all w K oe has w / for all ( b w Let q ax{ e : w K For all w K defie ( ( b j! I ( w e c ( j l! j l jl ( b ( w j l j l ( w The for all w K it follows fro Lea 6 ad ( that I ( w q b e ( b This iplies uifor covergece o K Step : We prove that for soe δ > the relatio i γ δρ ( O ( e cos γ ρ ± δw J e holds thus e J ( w O ( for arg w ± γ Proof step : Let γ be as i (6 The ( ρ a iγ b iγ iγ zρe zρe H e f ( z e dz f ( z e dz a a ( ( b j! ( j l! l jl iγ ( b ( ρe e cj iγ l j l ( ρe i Deote the ifiite series by T ( ρ e γ ad ote that ρ γ The fro Lea 6 ad ( it follows that i e γ si ρ cos si ( i γ ( b ρ γ T e O e B ρ γ (5 Sice f ( z is bouded o the segets [ a a ] ad [ a b ] by stadard calculatios oe gets a b aρ cosγ bρ cos γ Bρ si γ O( e e a (6 a The by choosig the path of itegratio as the reflectio i the real axis of that used i ( we get that (5 ad (6 hold i for as well Thus ρe γ cos cos si ( iγ aρ γ b ρ γ Bρ γ ( ρ ± H e O e e (7 Fro the defiitio of J ( w above oe deduces that where Q ( w is the etire fuctio defied as a b ( ( ( ( ( ( a a (8 J w G w f z e dz G w f z e dz Q w Note that Q ( w is also writte as ( ( b j! G( w e ( j l! j l jl ( b ( w cj l j l ( w (9

5 Sciece Joural of pplied Matheatics ad Statistics 5; 3(4: ( ( b j! e c ( w G ( w j l jl ( b ( w l j j l ( j l! (3 where k ( w w G ( w k (3 k Oe observes that cobiig (7 ad (7 gives for every ε > cos si ( ε cos ( si ε i a D b D B J ( ρ e ± γ O e ρ γ πρ γ ρ e ρ γ π ρ γ ρ Fro (8 oe also deduces that B πd b cot γ ad sice ε is arbitrarily sall this yields ( / cos si ( / cos ( i a D b J ( ρ e ± γ O e ρ γ πρ γ e ρ γ Relatio (6 iplies that πd si γ / 4a cosγ thus δ / 4 i( a b we have for Therefore ( cos iγ δρ γ J ( ρ e ± O e ± γ (3 δw e J ( w O( arg w Step 3: δ w We show that e J ( w is a fuctio of expoetial type bouded i the agle arg w γ I particular for real w this iplies that J ( w O ( e δw thus J ( O ( e δ ( δ / This evetually yields the relatio c O ( e which cotradicts relatio (5 So two poits ca be obtaied i two differet but the relatioship is boud (5 both should be equal to zero the there is either a sigle poit ad this copletes the proof of the theore Proof step 3: We will show that (3 holds i the agle Iw γ I order to do this first we prove that J ( w is a etire fuctio of expoetial type Fro (8 observe that it suffices to work with the fuctio Q ( w Cosider soe η > so that η < For every w C so that w S where S is the syste defied i ( we partitio the sequece { ito two sets as follows: ad The we write where Q ( w is defied as η ( w { : w > η B( w { : w Q( w Q ( w Q ( w B ( ( b j! Q ( w G( w e c ( j l! j l jl ( b ( w j l ( ( w j l w (33 Siilarly oe defies QB ( w Cosider ow Q ( w We reark that i this case the coditio w S plays o role Note that fro Lea 6 we deduce for ay ( w that j l jl ( ( b j! cj l j l ( w j l b (! Q ( w G( w e b e ( b w ( b ( w ad observe that the series is bouded above by the oe i ( w This iplies that Q ( w O( e κ for soe κ > Next we cosider QB ( w We ca also write it as Thus e c ( b j! ( w G ( w j jl ( b ( w l j l ( ( (! B w j l j l (34 where G ( w is defied i (3 Note that for ay pseudo-ultiplicity of b j Sice w S the oe gets j there is a b j so that a j ( ( j G( w G( w G ( w G( w 3 e ( j w b w thus ( j the b j (35

6 8 Naser bbasi ad Molood Gorji: Coversely Covergece Theore of Fabry Gap Fix soe > The fro Lea 4 we get j ( ( ( ψ a j a a ψ a a j j j j 3e 3e 3 e e ε with the last iequality valid sice < Oe also observes that a b 4 w sice B ( w j j Thus for all B ( w oe has G ( w G ( w e iplies that there are costats ' > ad '' > for ay w C S ad all B ( w oe has This so that 4 ε w '' w G ( w e ' (36 Next observe that for ay B ( w we have w ( l η l Cobiig this with (36 shows that QB ( w is bouded above by The fro Lea 6 we get that b j! j j l '' w ( b ( w η ( j l B( w j l ' e e c ( j l! η ( Note also that fro Lea 5 oe gets ad cobiig this with (37 gives η ( b e (37 '' w ( b w ( b QB ( w e ' e B( w η η χ χ ( η ε ε w ( ( e e Q ( w ' e e e b e B '' w ( b w ε w ( b B( w w with the series bouded above by the oe i ( This iplies that QB ( w O( e σ for soe σ > provided w S Sice Q ( w Q ( w QB ( w it follows that Q ( w O ( e νw for soe ν > provided w S But accordig to a al e e B b B b l i{ : Γ (38 Γ 3 3 S is the uio of o-overlappig disks whose radius teds to zero Sice Q ( w is a etire fuctio its axiu value over ay such closed disk is obtaied o the boudary w ll these iply that Q ( w O ( e ν for all w It the follows that J ( w is a etire fuctio of expoetial type Cobiig this result with relatio (3 ad a Phrage-Lidelof theore 7 it yields δw e J ( w O( arg w γ (39 I particular for real w this iplies that J ( w O ( e δw thus J ( O( e δ (4 Note also that fro (8 ad (9 oe deduces that J The fro c (! (4 k ( k k G we ca write c ( ( [ ]! k k [ ] G k (4! J ( ( ( (! (43 If we ow apply (9 ad (4 - (43 it yields for everyε > ( δ ε ( c O e (44

7 Sciece Joural of pplied Matheatics ad Statistics 5; 3(4: Sice is arbitrary we get that ( δ / ( c O e (45 ( δ / k ( ck O( e k But this cotradicts with relatio (5 ad this copletes the proof of our theore 4 Coclusio I this study we exaie a variatio of the coverse of Fabry Gap theore Polya's result shows that i soe sese Fabry's result is the best possible Perhaps the eleetary ad direct proof that etioed above ight be of soe iterest To do this a sequece with a series of ew build ad reorderig the call usig the covergece of three series * ** f f f obtai upper ad lower bouds d usig the Phrage-Lidelof theore ad we will achieve the desired result i this paper ckowledgeets The author gratefully ackowledges the help of Prof E Zikkos to iprove the origial versio of the paper Refereces [] bbasi N Gorji M O Covergece a Variatio of the Coverse of Fabry Gap Theore Sciece Joural of pplied Matheatics ad Statistics 3 ( ( [] Berestei C ad Gay Roger Coplex alysis ad Special Topics i Haroic alysis (New York Ic: Spriger-Verlag (995 [3] Blabert M ad Parvatha R Ultracovergece et sigualarites pour ue classe de series d expoetielles Uiversite de Greoble ales de l Istitut Fourier 9( ( [4] Blabert M ad Parvatha R Sur ue iegalite fodaetale et les sigualarites d ue foctio aalytique defiie par u eleet LC-dirichletie Uiversite de Greoble ales de l Istitut Fourier 33(4 ( [5] Berlad M O the covergeve ad sigularities of aalytic fuctios defied by E-Dirichletia eleets ales des Scieces Matheatiques du Quebec ((998 5 [6] Boas RP Jr Etire Fuctios (New York: cadeic Press (954 [7] Erdos P Note o the coverse of Fabry's Gap theore Tras er Math Soc 57 (945-4 [8] Polya G O coverse Gap theores Tras er Math Soc 5 ( [9] Levi B Ya Distributio of Zeros of Etire Fuctios (Providece RI: er Math Soc (964 [] Levi B Ya Lectures o Etire Fuctios (Providece RI: er Math Soc (996 [] Leviso N Gap ad Desity Theores erica Matheatical Society Colloquiu Publicatios Vol 6 (New York: er Math Soc (94 [] Madelbrojt S Dirichlet Series Priciples ad Methods (Dordrecht: D Reidel Publishig Co (97 pp x 66 [3] Valiro M G Sur les solutios des equatios differetielles lieaires d'ordre ifi et a coeffciets costats Ecole Nor Tras 3 46 ( [4] Zikkos E O a theore of Nora Leviso ad a variatio of the Fabry Gap theore Coplex Variables 5 (4 (5 9-55