Iteatioal Mathematical Foum, 5,, o. 43, - 3 Taylo Tasfomatios ito Mulatu Lemma Savaah State Uivesity Savaah, a 344, USA Lemmam@savstate.edu Abstact. Though out this pape, we assume that <. The Taylo tasfomatios T ae give by the positive, egula mati defied by = ( ) ( ) ( =,,,,... ) a The pupose of this pape is to study these matices as mappigs ito. The ecessay ad sufficiet coditios fo T to be is poved. The stegth of T i the settig is ivestigated. Also, It is show that evey T mati is taslative.. Itoductio ad Bacgoud. The Taylo tasfomatios ae studied as mappigs i [4]. So, it a atual questio to as if thee is a theoy fo T i the settig that paallels the theoy of T i the c c settig. The aswe is affimative, ad it poduces this pape.. Basic Notatios ad Defiitios. Let A = ( a ) be a ifiite mati defiig a sequece to sequece summability tasfomatio give by ( A) = = a (.)
M. Lemma ( A) whee deotes the th tem of the image sequece A. The sequece A is called the A-tasfom of the sequece. Let y be a comple umbe sequece. Thoughout this pape, we shall use the followig basic otatios. C = { set of all coveget sequeces} { } : ( ) fo some (, ) { :( ) ( ) fo some (,) } = y y = O y y o = = ( ) = { : } A y Ay Defiitio. If X ad Y ae comple umbe sequeces, the the mati A is called a X-Y mati if the image Au of u ude the tasfomatio A is i Y wheeve u is i X. Defiitio. The summability mati A is said to be taslative fo a sequece u i ( A ) povided that each of the sequeces Y u ad Zu is i ( A ), Y u, u, u,... Z =, u, u,.... whee = { } ad { } u 3 u Defiitio 3. The mati A is -stoge tha the mati B povided that ( ) ( ) B A. 3. The Mai Results. Ou fist theoem deals with the ecessay ad sufficiet coditios fo T to be a mati. Rema : Note that if, the Mp fo some p (,) Lemma. If T is a mati, the < Lemma. If <, the T is a mati.
Taylo tasfomatios ito 3 Poof. We will show that implies that that Mp fo some p,. M > ad ( ) T. Note that implies + T = ( ) = ( ) ( ) ( ) ( ) = + = + Mp p ( ) ( )( ) + = ( ) ( ) ( ) = + Mp p < Mp. Hece, T ad thus T is a mati. Theoem. T is a mati if ad oly if < Poof The theoem follows by Lemmas &. Rema. The fact that T is ot a mati fo > ca also be justified by the followig eample. Eample. Suppose that 3 = ad let be a sequece such that ( ) The we will show that the sequece T 3 is ot i ( ) = ( ) ( ) T 3 3 + = ( ) ( ) ( )( ) = + 3 4 4 =. 4 =.
4 M. Lemma ( ) 3 = = + = < ( ) ( ) ( ) ( ) 8. 4 4 (3.) Hece, it follows that T 3 is ot i. Thus, T 3 is ot a. Theoem. Suppose ad acsi. T a =. The mati implies that Poof. The theoem easily follows usig Theoem ad the iequality < acsi < fo Lemma 3. If T 3 a, the ( T ) cotais a bouded ( ocoveget) sequece. Poof. Let = ( ). We will show that ( ) < < T. Note that ( ) ( ) ( T = ) = + ( ) ( ) ( ) + = = (3.3) = + ( ) ( )( ) = ( ) ( ) ( ) = +
Taylo tasfomatios ito 5 ( ) <. Hece ( ) T if T a mati Lemma 4. If Poof. Let p 3 T a, the ( ) < <, = ( p), ad ( ) T cotais a ubouded sequece. p p< < p. We will show that ( T ). We have = ( ) ( ) ( ) T = + ( ) ( )( ) ( p) = + = + = p p ( ) ( )( ) = ( ) ( ) ( ) = + p p < ( s), whee ( ) s = p ( + p). Now the hypothesis that T a ( p ) p< implies that s <. Hece, it follows that ( ) T. Theoem 3. The T mati is stoge tha the idetity mati i the settig. Poof. The theoem follows by Lemmas 3 & 4. Lemma 5. Suppose A = [ a ] is a mati ad
6 M. Lemma mati such that a = fo <, m> p (both positive iteges); the ( p m ) ( ) A A 8], whee the itepetatio fo p A ad m A is as give i [ 3, p. Theoem 3. If T is a positive itege geate tha) Poof. Let. T is a m T T mati, m T is also a mati (fo m a mati implies that ( ) T m 5, we have ( ) ( ) ad hece it follows that ( T ) Thus, m T is a. By Lemma. The et mai esult suggests that the T mati is settig. taslative i the Theoem 4. Evey T mati is taslative Poof. Let ( ) () Y ( T) ( ) Z ( T ) T ad, Whee Y ad. The we will show that: Z ae as defiitio. Let us fist show () holds. (3.5) ( ) ( ) ( ) = TY = + ( ) ( ) = = ( ) = = + ( ) + + +
Taylo tasfomatios ito 7 ( ) + = + = + ( ) + = + = + whee A + B (3.6) ad ( ) + A = + = ( ) + B = + = + (3.7) Now if we show both A ad B ae i, the () holds. The coditio that that A follows fom the hypothesis that ( T ), ad B will be show as follows: Obseve that (3.9) B ( ) + + = = + ( ) + + = + t dt + = ( ) + + dt + t + = = ( ) + = t dt + t = + Let M = + t =.
8 M. Lemma Note that t < < ad ( ) 3.8, * ad 3.9 implies that T implies that M. Obseve that M B = ( ) (3.) Now M ad ( ) implies that B. Thus, () holds. Net we will show that () holds. ( ) = ( ) ( ) + = TZ ( ) + ( ) + + + = = + ( ) + = = (- ) (3.) + = + + ( ) + ( ) = = + = + ( ) + ( ) C + D whee ad ( ) + ( ) C = + = D = + + ( ) + ( ) = (3.) (3.3)
Taylo tasfomatios ito 9 By Theoem, the hypothesis that T emais oly to show D. Note that implies that C, hece thee + + D = + Note that t ( ) + ( ) ( ) ( ) = + = + t dt = ad ( ) 3.4 * ad 3.5 implies that + ( ) + = dt + t = ( ) + ( ) + tdt t + = = Let + ( ) P t = + = T implies that P. Obseve that (3.4) (3.5) D ( ) P = +. (3.6) Now P ad ( ) implies that D. Thus, () holds ad the theoem follows. ACKNOWLEDMENTS. I would also lie to tha Samea Mulatu, Abyssiia Mulatu, Lela Mie, Saa Asat, Sitayehu Joes, Kidus Feadu, Kiubel Seyoum, Babbi Melau ad Samuel Epheam fo thei geat suppot ad ecouagemet duig my wo o this eseach pape ad othes.
3 M. Lemma REFERENCES..H. Hady, Diveget Seies, Chelsa Publishig Compay, New Yo, 99.. R.E Powell ad S.M. Shah, Summability Theoy ad its Applicatios, Petice Hall of Idia, New Delhi, 988. 3. Buce Shawe ad Buce Watso, Boel s Methods of Summability Theoy ad Applicatios, Claedo Pess. Ofod, 994. 4. Mulatu Lemma, Taylo Tasfomatios ito w, the Southeast Asia Bulleti of Mathematics (4) 8: 99-34. 5. Mulatu Lemma, Taylo Tasfomatios ito, accepted ad soo to appea i the Southeast Asia Bulleti of Mathematics. Received: Jue, 9