ON ABSOLUTE NORMAL DOUBLE MATRIX SUMMABILITY METHODS. B. E. Rhoades Indiana University, USA

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1 GLASNIK MATEMATIČKI Vol , ON ABSOLUTE NORMAL DOUBLE MATRIX SUMMABILITY METHODS B. E. Rhoades Indiana University, USA Abstract. It is shown that, if T is a row finite nonnegative double summability matrix satisfying certain conditions, then T k summability is stronger than N, p n k summability, for k 1. As special summability methods T we consider weighted mean and double Cesáro, C, 1, 1, methods. 1. Introduction Let a mn be a doubly infinite series with partial sums s mn. Denote by T the doubly infinite matrix with entries t mnij, 0 i m, 0 j n. For any double sequence {u mn } we shall define 11 u mn u mn u m1,n u m,n1 u m1,n1. For a four-fold sequence, like t mnij, it will be understood that 11 operates only on the first two subscripts, and ij operates only on the last two subscripts. Associated with T are two matrices T and ˆT, where m n t mnij t mnµν, m, n, i, j 0, 1,..., µi νj and ˆt,,i,j 11 t,,i,j, m, n 1, 2,..., ˆt 00 t 00 t 00. We shall define m n 1.1 T mn t mnµν s µν. µ0 ν Mathematics Subject Classification. 40B05, 40G05, 40C05, 40D25. Key words and phrases. Absolute summability, weighted mean matrix, Cesáro matrix, inclusion theorems. 57

2 58 B. E. RHOADES Let a n be a given series with partial sums s n, C, α the Cesáro matrix of order α. If σn α denotes the n-th term of the C, α-transform of {s n }, then, from Flett [5], a n is summable C, α k, k 1, if 1.2 n k 1 σn α σ α k <. n1 An appropriate extension of 1.2 to arbitrary lower triangular matrices would be 1.3 n k 1 t k <, where n1 t n : n a ni s i, i0 and is the forward difference operator satisfying t n t n t n1. Such an extension is used in [3]. However, in [4], Bor and Thorpe replace 1.2 with the following definition. A series a n is said to be summable N, p n k, k 1 if Pn k Z k <, p n n1 where N, p denotes the weighted mean matrix generated by the sequence {p n }, with partial sums P n, and Z n : 1/P n n i0 p is i. Inequality 1.4 is apparently an attempt to extend 1.2 to weighted mean methods by interpreting n to be the reciprocal of the diagonal entries of C, 1. Unfortunately this is not an appropriate interpretation of 1.2. For, if it were, then, for the matrix methods C, α, one would have the condition n δ k 1 σn α σ α k <. n1 However, Flett [5, p. 115] continues to use 1.2. In spite of this fact, a number of papers have been published using 1.4. See, e.g., [1, 2, 4, 8]. In [7] the author proved the analog of the theorem in [8], using 1.3 instead of 1.4, and obtained some of the results of [1, 2, 3, 8] as corollaries. In this paper we obtain a two infinite-dimensional analog of absolute summability as defined in [7]. Adopting the two-dimensional analog of absolute summability as defined in [7], we shall say that the series a mn is absolutely T -summable of order k 1 if 1.5 mn k 1 11 T, k <. m1 n1

3 DOUBLE MATRIX SUMMABILITY 59 We shall define the mn-th term of the double weighted mean transform of a double sequence {s mn } by 1.6 Z mn 1 m n m n p ij s ij, P mn : p ij. P mn i0 j0 i0 j0 A doubly infinite weighted mean matrix will be called factorable if there exist sequences {p m }, {q n } such that p mn p m q n, and we focus on this case below. For any sequence {u ij }, 0j : u ij u i,j1, i 0 u ij : u ij u i1,j, and ij u ij : u ij u i,j1 u i1.j u i1.j1. 2. Main result Theorem 2.1. Suppose that a double factorable positive sequence {p mn } and a positive matrix T satisfy i t mnij t m1,n,i,j, t mnij t m,n1,i,j, m i, n j, i, j 0, 1,... ; ii P mn t mnmn Op mn ; m iii t mnµj t,n,µ,j an, j, µ0 and µ0 n t mniν t m,,i,ν bm, i, ν0 ν0 0 i m, 0 j n, m, n 0, 1,..., m µ0 ν0 n t mnµν 1, for m, n 0, 1,... ; iv 11 t,,i,j 0, for 0 i < m, 0 j < n, m, n 1, 2,... ; m n v t ijij ˆt,,i,j Ot mnmn ; vi vii viii ix mnt mnmn k 1 ˆt,,i,j Oijt ijij k 1 ; mi nj m n pi m mi nj 0jˆt,,i,j Ot mnmn ; P i n qj i0ˆt,,i,j Ot mnmn ; Q j mnt mnmn k 1 0jˆt,,i,j O ijt ijij k 1 q j Q j

4 60 B. E. RHOADES x xi m1 nj mi nj mnt mnmn k 1 i0ˆt,,i,j O ijt ijij k 1 p i P i mnt mnmn k 1 ijˆt,,i,j Oij k 1 t k ijij. If a mn is summable N, p mn k, then it is also summable T k, k 1. The notation 11 t,,r,s : t,,r,s t m,,r,s t,n,r,s t mnrs. Lemma 2.2. If T satisfies conditions i, iii, and iv, then T satisfies xii ˆt,,i,j 0 for 0 i m, 0 j n, m, n 1, 2,.... Proof. Using iii, for 0 i < m, 0 j < n, ˆt,,i,j t,,i,j t m,,i,j t,n,i,j t mnij m n t,,r,s t m,,r,s ri 1 0, sj m n ri sj r0 s0 1 1 t mnrs ri sj t,,r,s r0 sj m t m,,r,s r0 s0 r0 s0 r0 sj n t,n,r,s m n 1 t mnrs r0 s0 r0 s0 r0 sj r0 sj 11 t,,r,s s0 t mnrs ri sj t,,r,s t m,,r,s t,n,r,s t m,,m,s t mnms t,n,r,s 11 t,,r,s t,n,r,n t mnrn r0 sj since, by iv, the terms involving the double summation are nonnegative and, from i, t m,,m,s t mnms, and t,n,r,n t mnrn. r0

5 DOUBLE MATRIX SUMMABILITY 61 Suppose that 0 i < m, j n. Then, since t,,i,n t m,,i,n 0, and using i and iii, ˆt,,i,n t,n,i,n t mnin t,n,r,n ri an, n t,n,r,n an, n r0 t,n,r,n t mnrn 0. r0 Similarly, for i m, 0 j < n, ˆt,,m,j 0. Finally, ˆt,,m,n t mnmn 0. r0 m ri t mnrn t mnrn We remark that condition xii can be weakened. All that is required is that the ˆt,,i,j be of constant sign for all m and n. In condition iii one needs only that m n µ0 ν0 t mnµν c for some constant c. Proof of the Theorem 2.1. To carry out the proof we will solve 1.6 for a mn formula 2.2. Then, after using 1.1 to compute 11 T, 2.3, substitute 2.2 into 2.3 to obtain an expression for 11 T, in terms of Z mn. Then apply 1.5. From 1.6, since the weighted mean method is factorable, Z mn 1 m n p µν s µν 1 m n µ ν p µν a ij P mn P mn and 1 P mn 1 It then follows that m µ0 ν0 m n i0 j0 n m a ij i0 j0 m n µi νj Z mn Z m, Q Q n p µν µ0 ν0 i0 j0 n a ij P m P Q n Q a ij 1 P 1 Q, q n m n p m q n Z, P P m Q Q n a ij 1 P Q. P m m n a ij P Q,

6 62 B. E. RHOADES since P 1 Q 1 0. Solving 2.1 for a mn yields 2.2 From 1.1, and a mn P mq n p m q n 11 Z, P m 2Q n p q n 11 Z m 2, T mn P mq n 2 p m q 11 Z,n 2 P m 2Q n 2 p q 11 Z m 2,n 2. T m, m µ0 ν0 m i0 j0 n t mnµν s µν n a ij m µi νj m a ij t m,,i,j, i0 j0 i0 m µ0 ν0 n µ t mnµν i0 j0 ν i0 j0 a ij n m n t mnµν a ij t mnij. m m T mn T m, a in t mnin a ij t mnij t m,,i,j m i0 j0 i0 j0 n a ij t mnij t m,,i,j, since t m,,i,n 0. m n m T mn T m, a ij t mnij t m,,i,j a i0 t mni0 t m,,i,0 By iii, n a 0j t mn0j t,n,0,j. j1 t mni0 t m,,i,0 ri s0 i0 m n m t mnrs m n ri Similarly, t mn0j t,n,0,j 0. Thus T mn T m, m s0 ri s0 t mnrs s0 t m,,r,s t m,,r,s 0. n a ij t mnij t m,,i,j

7 DOUBLE MATRIX SUMMABILITY 63 and T, Substituting 2.2 into 2.3 yields 11 T, m n m n a ij 11 t,,i,j. ˆt,,i,j [ Pi Q j 11 Z, P iq j 2 11 Z,j 2 1 P i 2Q j 11 Z i 2, p q j P i 2Q j 2 11 Z ] i 2,j 2 p q Using the substitutions r i 1 in the second sum, s j 1 in the third sum, and both substitutions in the fourth sum, we have since 11 T, m n r1 j1 ˆt,,i,j P i Q j 11 Z, n m i1 s1 m r1 s1 n ˆt,,r1,j P r 1 Q j 11 Z r 1, p r q j ˆt,,i,s1 P i Q s 1 11 Z,s 1 p i q s ˆt,,r1,s1 P r 1 Q s 1 11 Z r 1,s 1 p r q s 1 11 Z, [P i Q jˆt,,i,j P Q jˆt,,i1,j P i Q ˆt,,i,j1 P Q ˆt,,i1,j1 ], ˆt,,m1,j ˆt,,i,n1 ˆt,,m1,j1 ˆt,,i1,n1 0, and P 1 Q 1 0.

8 64 B. E. RHOADES The quantity in brackets can be written in the form Q j [p iˆt,,i,j P ˆt,,i,j ˆt,,i1,j ] p i Q ˆt,,i,j1 P Q ˆt,,i,j1 ˆt,,i1,j1 ˆt,,i,j p i Q ˆt,,i,j ˆt,,i,j1 q j P ˆt,,i,j ˆt,,i1,j P Q ˆt,,i,j ˆt,,i1,j ˆt,,i,j1 ˆt,,i1,j1. Therefore 11 T, I 1 I 2 I 3 I 4, say. Substituting into 1.5 and using Hölder s inequality, we have J 1 : M m1 n1 M m1 n1 M m1 n1 N mn k 1 I 1 k N mn k 1 m n k ˆt,,i,j 11 Z, N mn k 1 Using v, vi, and 1.5, J 1 O1 O1 O1 O1 M m n m n t ijij ˆt,,i,j k 1. m1 n1 M M N mnt mnmn k 1 N M N t 1 k ijij ˆt,,i,j 11 Z, k m t 1 k ijij 11Z, k n M mi nj t 1 k ijij 11Z, k ijt ijij k 1 N ij k 1 11 Z, k O1. t 1 k ijij ˆt,,i,j 11 Z, k N mnt mnmn k 1 ˆt,,i,j

9 DOUBLE MATRIX SUMMABILITY 65 Using vii, ix and ii, J 2 : M m1 n1 M m1 n1 M m1 n1 O1 N mn k 1 I 2 k N mn k 1 m n N mn k 1 m n 11 Z, 0jˆt,,i,j q j Q 11 Z, k m n M m1 n1 N mnt mnmn k 1 Qj q j k Pi p i k 1 ojˆt,,i,j m n pi P i ojˆt,,i,j Qj q j k Pi p i k 1 k 1 ojˆt,,i,j 11 Z, k M N Qj k Pi k 1 M N O1 11 Z, k mnt mnmn k 1 q j p i mi nj 0jˆt,,i,j M N Qj k Pi k 1 O1 ijtijij k 1 q j 11 Z, k q j p i Q j O1 M N ij k 1 11 Z, k O1. k Using the same argument for J 2, and using the estimates viii, x, and ii, J 3 : M m1 n1 M m1 n1 O1. N mn k 1 I 3 k N mn k 1 m n P k i0ˆt,,i,j 11 Z, p i

10 66 B. E. RHOADES ˆt,,i,j t,,i,j t m,,i,j t,n,i,j t mnij Therefore and hence ri m m t,,r,s m sj n ri sj ri sj t mnrs n 11 t,,r,s. ri sj ˆt,,i,j ˆt,,i,j1 t,,r,s n ri sj m 11 t,,r,s, ri t,n,r,s 2.4 ijˆt,,i,j :ˆt,,i,j ˆt,,i1,j Using 2.4 and Hölder s inequality, J 4 : m m M m1 n1 M m1 n1 M m1 n1 N mn k 1 I 4 k N mn k 1 m n ˆt,,i,j1 ˆt,,i1,j1 11 t,,i,j. N mn k 1[ m n [ m n k t,,i,j ] P Q k ijˆt,,i,j 11 Z, Pi Q j k ij t,,i,j 11 Z, k] 11 t,,i,j 11 t,,m,n 11 t,,i,n j1 i1 m n 11 t,,m,j 11 t,,i,j M 1 M 2 M 3 M 4, say. M 1 t mnmn.

11 DOUBLE MATRIX SUMMABILITY 67 Using i and iii, and M 2 i1 t,n,i,n t mnin t,n,i,n t mnin i1 an, n t,n,0,n an, n t mn0n t mnmn t mnmn, M 3 t m,,m,j t mnmj t m,,m,j t mnmj j1 j1 bm, m t m,,m,0 bm, m t mnmn t mnm0 t mnmn, Using iv, iii, and 1, M 4 i1 11 t,,i,j j1 i0 i0 11 t,,i,j j0 t,,i,j t m,,i,j t,n,i,j t mnij j0 [bm 1, i bm, i bm 1, i t,n,i,n bm, i t mnin ] i0 an, n an, n t mnmn. Therefore, using xi and ii, M N J 4 O1 mnt mnmn k 1 m1 n1 m n ijˆt,,i,j 11 Z, k M N Pi Q k 11 j O1 Z, k O1 O1 M M mi nj N M Pi Q j k N mnt mnmn k 1 ijˆt,,i,j Pi Q j k 11 Z, k ij k 1 t ijij k N ij k 1 11 Z, k O1.

12 68 B. E. RHOADES 3. Applications We shall now use the Theorem to obtain some inclusion relations between weighted mean methods and double Cesáro matrices. Lemma 3.1. If T is a positive factorable weighted mean matrix, then conditions i and iii of Theorem 2.1 are automatically satisfied. Also 11 t,,i,j 0 for 0 i < m, 0 j < n, m, n 1, 2,.... and Proof. If T is a weighted mean matrix, then t mnij p ij p ij t m1,n,i,j P mn P m1,n Since t mnij p ij P m,n1 t m,n1,i,j. P mn m i0 j0 n p ij, condition iii is clearly satisfied, since T is factorable. For 0 i < m, 0 j < n, 11 t,,i,j t,,i,j t m,,i,j t,n,i,j t mnij p iq j p iq j p iq j P Q P m Q P Q m Q P P P m p m q n P P m Q Q n Note that 3.1 is the opposite of condition iv of Theorem 2.1. Corollary 3.2. Suppose that {p m }, {p m }, {q m }, {q m } are positive sequences such that i q n /Q n Oq n/q n, q n/q n Oq n /Q n, ii p m /P m Op m /P m, p m /P m Op m/p m, mn k 1 pm q k n ijpi q j k 1 iii O P mi nj Q P i Q j k. Then, if amn is summable N, p m, q n k, it is summable N, p m, q n k.

13 DOUBLE MATRIX SUMMABILITY 69 Proof. Suppose that a mn is summable N, p m q n k. Then, if we set T N, p m q n in the Theorem, from Lemma 3.1, conditions i and iii are automatically satisfied, and condition ii of the Theorem is implied by i and ii above. Condition iv of the Theorem is used to prove that 3.2 ˆt,,i,j 0. But, from 2.3 and 3.4 below, 3.2 is automatically satisfied. It remains to show that the remaining conditions of the Theorem are implied by the conditions of Corollary 3.2. We may write condition v of Theorem 2.1 as J 5 : t mnmn ˆt,,m,n t inin ˆt,,i,n By property iii 3.3 Therefore J 52 i1 t mjmj ˆt,,m,j j1 J 51 J 52 J 53 J 54, say. i1 j1 J 51 t 2 mnmn t mnmn. ˆt,,i,n t,,i,n t mnin m t,n,r,n i1 p i q n P i Q n ri ri t mnin t ijij ˆt,,i,j an, n t,,r,n an, n r0 r0 r0 pr q n p rq n p mp q n. P Q n P m P Q n p m P q n P m P Q n Similarly, J 53 Ot mnmn. t,,i,j ri p mq n 1 P mq n P t,,i,j sj i1 ri t mnin p i p mq n t mnmn. sj P P Q Q. Q P P Q

14 70 B. E. RHOADES ˆt,,i,j t,,i,j t m,,i,j t,n,i,j t mnij 1 P 1 Q 1 P 1 Q P Q P m Q 1 P 1 Q 1 P 1 Q P Q n 1 Q 1 P 1 P Q P P m 1 Q 1 P 1 P Q n P P m 1 Q P p m 1 Q P p m Q P m P Q P m P p mp 1 Q 1 Q P m P Q Q n 3.4 p mq n P Q P m P Q n Q. Therefore J 54 i1 P i Q j j1 p m q n P Q P m P Q n Q p m q n P m P Q m Q P Q p mq n t mnmn, and condition v is satisfied. Using 3.4 and condition ii of Corollary 3.2, mi nj mnpm q n P m P Q n Q k 1 p m q n P Q P Q mi nj mn k 1 P Q pm q n k ijpi q j k 1 P Q O P i Q j k Oijt ijij k 1, and condition vi is satisfied.

15 DOUBLE MATRIX SUMMABILITY 71 Using condition ii of Corollary 3.2 we may write condition vii of the Theorem as [ pm O1 0nˆt,,m,n P m j1 pm i1 0jˆt,,m,j P m O1[J 71 J 72 J 73 J 74 ], say. pi P i 0nˆt,,i,n i1 j1 pi ] 0jˆt,,i,j P i Using 3.3, 0nˆt,,i,n ˆt,,i,n ˆt,,i,n1 ˆt,,i,n. J 71 p m P m t mnmn t mnmn, and J 72 i1 pi P i pm P q n P m P Q n p mq n t mnmn. Similarly, J 73 t mnmn. Using 3.4, Therefore 0jˆt,,i,j ˆt,,i,j ˆt,,i,j1 J 74 p mq n P Q p mq n P Q j P m P Q n Q P m P Q n Q p mq n P q j P m P Q n Q. i1 pi j1 P i p m q n P m P Q n Q p m q n P m P Q n Q P Q p mq n t mnmn, and condition vii is satisfied. The validity of condition viii of Theorem 2.1 is proved similarly.

16 72 B. E. RHOADES Using iii, i, and ii of Corollary 3.2, condition ix becomes mnpm q k 1 n p m q n P q j P m P Q n Q mi nj P q j mi nj ijpi q j k 1 P q j O P i Q j k O ijt ijij k 1 q j O Q j mn k 1 P Q pm q n k Condition x is proved in a similar manner. Using 3.4, ijˆt,,i,j 11 t,,i,j ijpi O P i k 1 qj Q j k ijt ijij k 1 q j Q j t,,i,j t m,,i,j t,n,i,j t mnij p iq j p iq j p iq j P Q P m Q P Q n [ ] Q P P P m p m q n. P m P Q n Q Therefore, using iii of Corollary 3.2, mnpm q k 1 n p m q n P m P Q n Q mi nj mi nj ijpi q j k 1 O P i Q j k Oij k 1 t k ijij, and condition xi is satisfied. mn k 1 P Q pm q n k. O ij k 1 p i q k j P i Q j Corollary 3.3. Let {p m }, {q n } be positive sequences satisfying i mp m OP m, P m Omp m, ii nq n OQ n, Q n Onq n. Then a mn is summable C, 1, 1 k if and only if it is also summable N, p m q n k, k 1.

17 DOUBLE MATRIX SUMMABILITY 73 Proof. First apply Corollary 3.2 with p m q n 1 for all n. Conditions i and ii of Corollary 3.3 imply conditions i and ii of Corollary 3.2. Using conditions i and ii of Corollary 3.3, mn k 1 pm q k n P Q mi nj O1 mi nj p m q n O1 O P m P Q n Q P i Q j ij k 1 P i Q j k, and condition iii of Corollary 3.2 is satisfied. With T C, 1, 1, i.e., p m q n 1 for each m and n, condition iii of Corollary 3.2 becomes mi nj 1 mm 1nn 1 1 ij O ij k 1 P i Q j k. References [1] H. Bor, On two summability methods, Math. Proc. Cambridge Phil. Soc , [2] H. Bor, A note on two summability methods, Proc. Amer. Math. Soc , [3] H. Bor, On the relative strength of two absolute summability methods, Proc. Amer. Math. Soc , [4] H. Bor and B. Thorpe, A note on two absolute summability methods, Analysis , 1 3. [5] T.M. Flett, On an extension of absolute summability and theorems of Littlewood and Paley, Proc. London Math. Soc , [6] B.E. Rhoades, Absolute comparison theorems for double weighted mean and double Cesáro means, Math. Slovaca , [7] B.E. Rhoades, Inclusion theorems for absolute matrix summability methods, J. Math. Anal. Appl , [8] M. Sarigöl, On absolute normal matrix summability methods, Glasnik Mat , B. E. Rhoades Department of Mathematics Indiana University Bloomington, IN USA rhoades@indiana.edu Received: Revised: &

Note on Absolute Matrix Summability Factors

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