Fibonacci Matrix Summability of Fourier series

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1 Ieraioal Joural o Mahemaics ad Saisics Ieio (IJMSI) E-ISSN: P-ISSN: Volume 5 Issue 8 Ocober. 7 PP-3-36 iboacci Marix Summabiliy o ourier series *Ahmadu Kilho, Abdullahi Mohammed ad Ado Balili *Deparme o Mahemaical Scieces, Uiersiy o Maiduguri, Boro Sae, Nigeria ABSTRACT: This oe aims a applyig recely deied iiie marix mehod which we apply i he summaio o ourier series. The summaio amous o uiorm coergece o ourier series as maiaied by ejer, sice94. Keywords: iboacci marix, ourier series, Marix mehod, Uiorm coergece, Mahemaical Subjec Classiicaio: 4A5,4C5, 4D Dae o Submissio: 3--7 Dae o accepace: I. INTRODUCTION I 3, Kalma ad Mea wroe ha amog umerical sequeces he iboacci umbers achieed a kid o celebriy saus, because hey are amous or possessig woderul ad amazig properies some are well kow. Amog hese properies, are ha he dierece o wo iboacci umbers is a iboacci umber, raios o iboacci umbers coerge o he golde mea, ay our cosecuie iboacci umbers are iboacci umbers, he greaes commo diisor, gcd, o wo iboacci umbers is aoher iboacci umber, jus o meio bu ew. So, iboacci umbers sad ou o be a kid o super sequece. To sar wih, iboacci sequece say, are he erms o he sequece,,,, 3, 5,... wherei each erm is he sum o he wo precedig erms sarig wih ad deoed by ad, respeciely. The ame iboacci sequeces is due o racois Edwouard Aaole Lucas i 876. Sum o squares, asympoic behaior, ruig sums, iie marix orm o iboacci umbers ca be see i [5]. Recely, iiie marices geeraed by iboacci umbers has bee used i he works o [6]. The book by [8], ersio I eel, is he pricipal auhoriy i he sudies o ourier series, wherei he also wroe a cerai orm o rigoomerical series called ourier series o orm: a ( c o s s i ) a x b x where he coeicies a, a,..., b, b,... are idepede o x. The coeicies are real, ad sice all he erms o () are o period, i is suicie o sudy rigoomerical series i ay ieral o legh, or example, (, ) o r (, ). The sysem o ucios, co s x, si x, co s x, si x,..., called he rigoomerical sysem is orhogoal i (, ). urhermore, le ' I s i m x s i x a d I, I m, m, m, co s m x si x a d co s m x co s x. Iegraig he ormula '' deoig he correspodig iegrals wih si m x si x co s( m ) x co s( m ) x ad akig io accou he periodiciy o iegral ucios, he we id ha I w h e e e r m. This is rue ee whe m =. The ' s are ow, m,,,..., ad so, i or a gie ucio we pu a ( ) co s d, b ( ) si d () The () is called he ourier series o. O chagig he deiiio o ourier series i he case o a shall call a ad b he ourier coeicies o he ucio. So may works o he sudy o ourier series ad heir coergece were carried ou by seeral auhors sarig rom [5] who worked maily o ourier series. This paper cao coai all he reereces o research works i his respec sarig rom lae 8 s. Howeer, he ollowig periodical papers as grouped below were eough or his piece o work: (), we 3 Page

2 iboacci Marix Summabiliy o ourier series O summabiliy ad coergece o ourier series see [], [3], [4], [6] ad [7]; o Cesaro summabiliy o ourier series see [6], [7], [8] ad [9]; o liear ad riagular mehods o summabiliy see [] ad []; ad o marix mehods o summabiliy see [3] ad [4]. I 9, Teopliz, O. published some work cocerig iiie marices wih codiios. They were marices o he orm: a a a... a... a a a... a... A a a a... a The marix A is called a Teopliz marix or T-marix i (i) lim a,,,,..., i i (ii) lim A ad i i (iii) N C, i,,,... i where C is idepede o i, see ([8], p. 39). Summabiliy is abou geeralizaio o he coergece o sequeces ad series. The use o iiie marices o realize coergece cao be oer emphasized. Ay iiie marix used i summabiliy o sequeces ad series is called a mehod o summabiliy, or isace he (C, ) mehod called he Cesaro mehod o summabiliy, where he limi o a sequece (x ) accordig o Cesaro ca be deied as limy, where (y ) is gie by y i p x i. I [5] ejer s heorem o he oher had saes ha i a ucio L (, ), p he he Cesaro s meas y o he parial sum o he ourier series o p coerges o i he L - orm. I i addiio, is coiuous ad ( ) ( ), he y coerges uiormly o. Seeral geeralizaios o (C, ) mehod led o he mehods (C, k), up o (C, ). The work o Bosaque, see [] is regarded as he bes geeralizaio o (C, ) mehod. We remark ha i is aural replace o replace (C, ) mehod wih ay oher mehods. We wish o replace wih iboacci iiie marix deoed by, deied i [6] as ollows:, o r ( ),, o r I is a riagle, ha is, a d o r (,, 3,... ). I is also a regular marix, or i saisies he codiio o regular marices as spel ou i [9], ha ay marix A ( a ), is regular i ad oly i ( i) h ere ex iss M, such ha or eery,, 3,... ( ii) lim a o r eery,,... ad ( iii ) lim a. a Clearly, he iboacci marix is regular. We wish o esablish a aalogue ha is replaced by (C, ) mehod is replaced, he rasorm o he o he parial sum o he ourier series o -period coiuous ucio coerges uiormly o. To do his, we eed o ix oaios ad some ecessary precedig lemmas o he mai resul o his paper. (3) 3 Page

3 II. BASIC DEINITIONS AND LEMMAS iboacci Marix Summabiliy o ourier series Deiiio. C { : is p e rio d ic a d c o i u o u s}, he se o all coiuous, - periodic ucios. I ay is -periodic ad (, ) is Riema iegrable, he () holds. I, h e b ad he ourier series () is he ourier series o wih ourier coeicies a a d b.or each C a d he equaliy where, s (, ) ( x ) D ( x ) d x si ( ) ( ) D,, is called he Drichle kerel. Ad s (, ) he h si parial sum o he ourier series (). D has he ollowig properies: Lemma.: ( ) C o r all, a d ( ) ( ) or ay a d. I he equaio: (, ) s (, ) ( x ) K ( x ) d x, w ih K ( ) a D ( ) (4) (, ) is he sequece o parial sums o he ourier series o ay periodic coiuous ucio, while K ( ) is he h kerel correspodig o he iboacci marix,. We wish o show ha (, ), he rasorm o s (, ) uder coerges uiormly o. We wish o adop he ollowig oaios: Le ( x ; ) ( x ) K ( ) d (5) ( x ) K ( ) d ( x ) K ( ) d (6) Noe ha K ( ) K ( ), o r h e k e r e l K ( ) is periodic ad coiuous ucio. Also, ( x ) ( x ) ( x ; ) K ( ) d (7) ( x ) lim ( x ) ( x ) (8 ) This ucio is deied a hose pois or which he precedig limi exiss. I is coiuous a some poi x he ( x ) ( x ) ; ad i i has a jump discoiuiy a x, he ( x ) ( x ) ( x ). Lemma. (Kalma ad Mea, [5]): Le ( ) a d a s Lemma.3 (Kara ad Basarir, [6]): Le ( ) ( i ), coerges, ad ( ii ) s u p ( ) be a sequece o iboacci umbers. The heir sum o squares be a sequece o iboacci umbers. The 3 Page

4 Lemma.4: or ay bouded sequece such ha x M,, we hae x x M iboacci Marix Summabiliy o ourier series M. Lemma.5 ([7] p.44): Le h kerel o a ideiy iiie marix be deoed by K ( ) saisyig he ollowig codiios: ( a ) K ( ) d, ( b ) K ( ), ( c ) K ( ) d C wih C idepede o, ad ( d ) i ( ) m ax K ( ), ( ) he ( ) o r each ix ed. The he h kerel is posiie i i saisies codiios (a) ad (b); ad is called quasi-posiie i i saisies codiio (c) oly. Lemma.6 ([8], p.46): (i) I K ( ) is a posiie kerel, he or ay saisyig m M, we hae m ( x, ) M. (ii) I K ( ) is a quasi-posiie kerel, ad M he i implies ha ( x, ) C M, wih C as i codiio (c). The las Lemma ad equaio (4) aboe sugges ha we mus perorm he calculaios or K ( ), he h kerel correspodig o he iboacci marix, deied i (3),ad subsequely or s u p L ad ( ) m ax K ( ). So, we should hae: where, K ( ) a D D ( ) D s i ( ) s i D si ( ) K ( ), si si ( ), si 33 Page

5 iboacci Marix Summabiliy o ourier series s i ( ) s i s i s i ( ) s i (c o s c o s ( )) s i s i ( c o s ( ) s i ( ) o r, K ( ) s i s i ( ) K ( ) s i Also, s i s i ( ) ( ) m a x K ( ), s i, o r III. MAIN RESULT Theorem 3.:Le C [, ] be such ha s (, x ) s (, x ) o r, x ( ) saisyig ( i ) ( ), (ii), (iii) su p L K ( ) d, ad (9) ad a marix 34 Page

6 iboacci Marix Summabiliy o ourier series (i) ( ) m ax K ( ), or all (,). The C [, ],, we hae: (, x) ( x ), uiormly. Proo: Assume ha ( x ) exiss a some poi x [, ]. The here exiss some such ha ( x ) ( x ) ( x ), or all. I iew o uiorm coiuiy o, we ca selec a o be idepede o x [ a, b ]. So, rom (5) ad (7) we hae, ( x ) ( x ) ( x ) ( x ) K ( ) d ( x ) ( x ) ( x ) K ( ) d ( x ) ( x ) ( x ) K ( ) d Le us perorm some ecessary esimaes as ollows: Ad, sice This implies ha K ( x ) ( x ) K K ( ) d s i ( ) ( ) 4 s i ( x ) K ( ) d ( ) d ( ), o r e a c h. 4 s i ( x ) ( x ) H ( x ) ( x ) d ( ) 4 4 s i where, H ( x ) ( x ) ( x ) si d Clearly, H is idepede o x. So, or some N, choose H, he he esimaes i () ad () yield 4 ( ), is rue or all N ad or all x [ a, b ] proided is coiuous o x x [, ]. Thus ( ), uiormly. Q.E.D. 35 Page

7 iboacci Marix Summabiliy o ourier series IV. CONCLUSION I coclusio we hae successiely used iboacci Marix o rece origi o realize uiorm coergece o ourier series, a mehod diere rom ejér sice 94. Thus he coribuio is ew mehod, or proo, o show uiorm coergece o ourier series. ACKNOWLEDGEMENT We would like o express our graiude o he reiewer or his/her careul readig ad aluable suggesios which improed he represeaio o he paper. REERENCES []. L. S. Bosaque, O he sudy o ourier series, Proceedigs o Lodo Mahemaical Sociey, (3) 93, []. E. Hille, ad J. D. Tamarki, O he sudy o ourier series, Trasacios o America Mahemaical Sociey, (34) 93, [3]. O Szasz, Coergece properies o ourier series, Trasacios o America Mahemaical Sociey, 934, [4]. L. S. Bosaque, Noe o ourier series, Composio Mahemaica, () 935, 8 87 [5]. S. Izumi, Some rigoomerical series VI, Tohoku Joural o Mahemaics, 5 (3) 954,9 95 [6]. L. ejér, Uersuchuge über ouriesche, Mahemaische Aale, 94, 5 69 [7]. S. Yao, Noe o ourier aalysis XXXI: Cesaro summabiliy o ourier series, Paciic Joural o Mahemaics,() 95, [8]. S. Yao, Cesaro summabiliy o ourier series, Tohoku Joural o Mahemaics, 5 (3) 953, [9]. K. Kao O he Cesaro summabiliy o ourier series (II), Tohoku Joural o Mahemaics, (7) 955, 8 []. K. Yao, O a mehod o Cesaro summaio o ourier series, Kodai Mah. Sem.Rep.. 9 () 956, []. A. V Eimo, O liear summabiliy mehods or ourier series, Akad. NaukSSSR Ser. Ma.. 4, (5) 96, []. H. P. Dikshi,, Summabiliy o ourier series by riagular marix rasormaios, Paciic Joural o Mahemaics, 3, () 969, [3]. A. M.Jarrah, ad E. Malkowsky, Ordiary, absolue ad srog summabiliy ad marix rasormaios, iloma, (7) 3, [4]. G. H. Hardy, Dierge series, Oxord Uiersiy Press, Lodo, 949 [5]. D. Kalma, ad R. Mea, The iboacci umbers-exposed, Mahemaics Magazie,76, (3) 3, 67 8 [6]. E. E. Kara, ad M. Basarir, A applicaio o iboacci umbers io iiie Toepliz marices, Caspia Joural o Mahemaical Scieces, (), [7]. A. Zygmud, A remark o cojugae series, Proceedigs o Lodo Mahemaical Sociey, (3) 93, [8]. A. Zygmud, Trigoomerical series, Doer Publicaios, USA, 955 [9]. A. Wilasky, Summabiliy hrough ucioal Aalysis, Mah. Sudies, Norh - Hollad, 984 Ahmadu Kilho "iboacci Marix Summabiliy o ourier series. Ieraioal Joural o Mahemaics ad Saisics eio(ijmsi), ol. 5, o. 8, 7, pp Page