LOGARITHMIC MATRIX TRANSFORMATIONS INTO G w

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1 IJMMS 2004:25, PII. S Hindawi Publishing Corp. LOGARITHMIC MATRIX TRANSFORMATIONS INTO G w MULATU LEMMA Received 0 March 2003 and in revised form July 2003 We introduced the logarithmic matrix L t and studied it as mappings into l and G in 998 and 2000, respectively. In this paper, we study L t as mappings into G w Mathematics Subject Classification: 40A05, 40C05.. Introduction. The logarithmic power series method of summability [], denoted by L, is the following sequence-to-function transformation: if { } lim x log( x) k+ u kx k+ = A, (.) then u is L-summable to A. The matrix analogue of the L-summability method is the L t matrix [2] givenby a nk t n k+ tk+ n, (.2) where 0 <t n < for all n and lim n t n =. Thus, the sequence u is transformed into the sequence L t u whose nth term is given by ( Lt u ) n log ( t n ) k+ u ktn k+. (.3) The L t matrix is called the logarithmic matrix. Throughout this paper, t will denote such a sequence: 0 <t n < foralln, and lim n t n =. 2. Basic notations and definitions. Let A = (a nk ) be an infinite matrix defining a sequence-to-sequence summability transformation given by ( Ax )n = a nk x k, (2.) where (Ax) n denotes the nth term of the image sequence Ax.ThesequenceAx is called the A-transform of the sequence x. Let y be a complex number sequence. Throughout

2 30 MULATU LEMMA this paper, we will use the following basic notations and definitions: { l = y : yk is convergent }, l(a) ={y : Ay l}, G = { y : y k = O ( r k) for some r (0,) }, G w = { y : y k = O ( r k) for some r (0,w), 0 <w< }, G w (A) = { } y : Ay G w, c ={the set of all convergent sequences}, c(a) = { y : A(y) c }. (2.2) Definition 2.. If X and Y are complex number sequences, then the matrix A is called an X-Y matrix if the image Au of u under the transformation A is in Y whenever u is in X. Definition 2.2. The summability matrix A is said to be G w -translative for a sequence u in G w (A) provided that each of the sequences T u and S u is in G w (A), where T u ={u,u 2,u 3,...} and S u ={0,u 0,u,...}. Definition 2.3. G w (A). The matrix A is G w -stronger than the matrix B provided that G w (B) 3. Main results. Our first main result gives a necessary and sufficient condition for L t to be G w -G w. Theorem 3.. t) G w. The logarithmic matrix L t is a G w -G w matrix if and only if /log( Proof. Since 0 <t n <, it follows that ank log ( t n ), (3.) for all n and k. Therefore, if /log( t) G w,[3, Theorem 2.3] guarantees that L t is a G w -G w matrix. Conversely, if /log( t) G w, then the first column of L t is not in G w because a n,0 t n /log( t n ) G w.hence,l t is not a G w -G w matrix by [3, Theorem 2.3]. Corollary 3.2. matrix. If 0 <t n <u n < and L t is a G w -G w matrix, then L u is also a G w -G w Corollary 3.3. Suppose α> and L t is an G w -G w matrix, then ( t) α+ G w. Corollary 3.4. only if r>/w. Corollary 3.5. Let t n = e qn, where q n = r n.thenl t is a G w -G w matrix if and If L t is a G w -G w matrix, then it is a G-G matrix.

3 LOGARITHMIC TRANSFORMATIONS INTO G w 3 The next result suggests that the logarithmic matrix L t is G w -stronger than the identity matrix. The result indicates that the L t matrix is rather a strong method in the G w -G w setting. Theorem 3.6. If L t is a G w -G w matrix and the series x k has bounded partial sums, then it follows that x G w (L t ). Proof. The proof easily follows using the same techniques as in the proof of Theorem 3.0 [2]. Remark 3.7. Theorem 3.6 indicates that if L t is a G w -G w matrix, then G w (L t ) contains the class of all conditionally convergent series. This suggests how large the size of G w (L t ) is. In fact, we can give a further indication of the size of G w (L t ) by showing that if L t is a G w -G w matrix, then G w (L t ) contains also an unbounded sequence. To see this, consider the sequence x given by Then Hence, x k = ( ) k (k+) 2 (k+2)(k+3). (3.2) k+ x ktn k+ = t n ( ) k (k+)(k+2)(k+3)tn k 6t n = ( ) 4. (3.3) +tn ( L t x ) 6t n n = log ( 6 )( ) 4 t n +tn. (3.4) t n Thus, if L t is an G w -G w matrix, then by Theorem 3., /log( t) G w,sox G w (L t ). The next few results deal with the G w -translativity of L t. We will show that the L t matrix is G w -translative for some sequences in G w (L t ). Proposition 3.8. G w. Every G w -G w L t matrix is G w -translative for each sequence x Theorem 3.9. Suppose L t is a G w -G w matrix and {x k /k} is a sequence such that x k /k = 0 for k = 0, then the sequence {x k /k} is in G w (L t ) for each L-summable sequence x. Proof. where Let Y be the L t -transform of the sequence {x k /k}. Then we have Yn t n k(k+) x ktn k+ C n +D n, (3.5) xtn C n 2log ( x2tn ) t n 6, t n D n (3.6) t n k(k+) x ktn k.

4 32 MULATU LEMMA By Theorem 3., the hypothesis that L t is G w -G w implies that C G w, and hence there remains only to show D G w to prove the theorem. Note that D n ( x tn k t dt) t n k (k+) 0 ( tn ) dt t n (k+) x kt k. 0 (3.7) The interchanging of the integral and the summation is legitimate as the radius of convergence of the power series k+ x kt k (3.8) is at least by [2, Lemma ] and hence the power series converges absolutely and uniformly for 0 t t n. Now we let = k+ x kt k. (3.9) Then we have log( t) log( t) k+ x kt k, (3.0) and the hypothesis that x c(l) implies that We also have lim = A(finite), for 0 <t<. (3.) t log( t) lim t 0 = 0. (3.2) log( t) Now (3.) and (3.2) yield and hence log( t) M, for some M>0, (3.3) M log( t), for 0 <t<. (3.4)

5 LOGARITHMIC TRANSFORMATIONS INTO G w 33 So, we have D n t n t n tn 0 tn 0 tn dt dt M log( t)dt t n 0 M ( ) Mt n t n t n M. t n (3.5) The hypothesis that L t is G w -G w implies that both /log( t) and ( t) are in G w by Theorem 3.. HenceD G w. Theorem 3.0. Every G w -G w L t matrix is G w -translative for each L-summable sequence in G w (L t ). Proof. Let x c(l) G w (L t ). Then we will show that (i) T x G w (L t ), (ii) S x G w (L t ). We first show that (i) holds. Note that ( L t T x ) n t n k+ x k+tn k+ t n k x ktn k k= ( ) t n k+ + x k tn k k(k+) k= P n +Q n, (3.6) where P n t n k+ x ktn k, k= Q n t n k(k+) x ktn k. k= (3.7) So, we have (L t T x ) n P n + Q n, and if we show that both P and Q are in G w,then (i) holds. But the condition P G w follows from the hypothesis that x G w (L t ) and Q G w follows from Theorem 3.9.

6 34 MULATU LEMMA Next we will show that (ii) holds. We have ( L t S x ) n t n k+ x k tn k+ k= t n k+2 x ktn k+2 W n +U n, (3.8) where W n t n k+ x ktn k+2, U n t n (k+)(k+2) x ktn k+2. (3.9) The hypothesis that X G w (L t ) implies that W G w. We can also show that U G w by making a slight modification in the proof of Theorem 3.9, replacing the sequence {x k /k} with the sequence {x k /(k+2)}. Hence, the theorem follows. Acknowledgments. I would like to thank the referee for great assistance. I want to thank my wife Mrs. Tsehaye Dejene Habtemariam and my two great daughters Samera Mulatu and Abyssinia Mulatu for their great cooperation during my work on this paper. I also want to thank Dr. Harpal Sengh, the Chair of the Department of Natural Sciences and Mathematics, for his great support and encouragement. References [] D. Borwein, A logarithmic method of summability,j.londonmath.soc.33 (958), [2] M. Lemma, Logarithmic transformations into l, Rocky Mountain J. Math. 28 (998), no., [3] S. Selvaraj, Matrix summability of classes of geometric sequences, Rocky Mountain J. Math. 22 (992), no. 2, Mulatu Lemma: Department of Natural Sciences and Mathematics, Savannah State University, Savannah, GA 3406, USA address: lemmam@savstate.edu

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