Abstract:

In this paper, we establish a summability factor theorem for
summability [[absolute value of A, [delta]].sub.k]. This paper is an
extension of the main result of Savas, E. A study on absolute
summability factors for a triangular matrix.

Key words: absolute summability, summability factors, triangular matrix.

Key words: absolute summability, summability factors, triangular matrix.

Article Type:

Report

Subject:

Matrices
(Research)

Author:

Savas, Ekrem

Pub Date:

06/01/2011

Publication:

Name: Proceedings of the Estonian Academy of Sciences Publisher: Estonian Academy Publishers Audience: Academic Format: Magazine/Journal Subject: Chemistry Copyright: COPYRIGHT 2011 Estonian Academy Publishers ISSN: 1736-6046

Issue:

Date: June, 2011 Source Volume: 60 Source Issue: 2

Topic:

Event Code: 310 Science & research

Geographic:

Geographic Scope: Turkey Geographic Code: 7TURK Turkey

Accession Number:

258241634

Full Text:

1. INTRODUCTION

Savas [2] obtained sufficient conditions for [summation][a.sub.n][[lambda].sub.n] to be [[absolute value of A].sub.k]- summable, k [member of] N. In this paper, a theorem on [[absolute value of A, [delta]]-summability methods is proved. This theorem includes a known result.

Let A be a lower triangular matrix, {[s.sub.n]} a sequence. Then

[A.sub.n] : = [n.summation over (v=0)] [a.sub.nv][s.sub.v].

A series [summation][a.sub.n] is said to be summable [[absolute value of A].sub.k], k [greater than or equal to] 1 if

[[infinity].summation over (n=1)][n.sub.k-1][[absolute value of [A.sub.n] - [A.sub.n-1]].sup.k] < [infinity], (1)

and it is said to be summable [[absolute value of A, [delta]].sub.k] k [greater than or equal to] 1 and [delta] [greater than or equal to] 0 if (see, [1])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

We may associate with A two lower triangular matrices [bar.A] and [??] defined as follows:

[[bar.a].sub.nv] = [n.summation over (r=v)] [a.sub.nr], n, v = 0, 1, 2, ...,

and

[[??].sub.nv] = [[bar.n].sub.nv] - [[bar.a].sub.n-1,v], n = 1, 2, 3,....

Also we shall define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

Given any sequence {[x.sub.n]}, the notation [x.sub.n] [??] O(1) means [x.sub.n] = O(1) and 1/[x.sub.n] = O(1). For any matrix entry [a.sub.nv], [[DELTA].sub.v][a.sub.nv] := [a.sub.nv] - [a.sub.nv+1].

2. MAIN RESULT

Theorem 1. Let A be a lower triangular matrix with nonnegative entries such that

(i) [[bar.a].sub.n0] = 1, n = 0, 1, ...,

(ii) [a.sub.n-1,v] [greater than or equal to] [a.sub.nv] for n [greater than or equal to] v + 1,

(iii) [na.sub.nn] [??] O(1), and

(iv) [[summation].sup.n=1.sub.v=1] [a.sub.vv][absolute value of [[??].sub.n,v+1]] = O([a.sub.nn]).

(v) [m+1.summation over (n=v+1)] [n.sup.[delta]k][absolute value of [[DELTA].sub.v][[??].sub.nv]] = O([v.sup.[delta]k] [a.sub.vv]) and

(vi) [m+1.summation over (n=v+1)] [n.sup.[delta]k][[??].sub.nv+1]] = O([v.sup.[delta]k]).

Let [t.sup.1.sub.n] denote the nth (C, 1) mean of {[na.sub.n]}. If

(vii) [[infinity].summation over (v=1)] [v.sup.[delta]k][a.sub.vv][[absolute value of [[lambda].sub.v]].sup.k] [[absolute value of [t.sup.1.sub.v]].sup.k] = O(1),

(viii) [[infinity].summation over (v=1)] [v.sup.[delta]k][absolute value of [a.sub.vv]].sup.1-k][[absolute value of [DELTA][[lambda].sub.v]].sup.k] [[absolute value of [t.sup.1.sub.v]].sup.k] = O(1),

(ix) [[infinity].summation over (v=1)] [v.sup.[delta]k][a.sub.vv][[absolute value of [[lambda].sub.v+1]].sup.k] [[absolute value of [t.sup.1.sub.v]].sup.k] = O(1),

then the series [summation][a.sub.n][[lambda].sub.n] is summable [[absolute value of A, [delta]].sub.k], k [greater than or equal to] 1, 0 [less than or equal to] [delta] < 1/k.

Proof. From (i) it follows that [[??].sub.n,0] = 0. Also ([[??].sub.n1] [[lambda].sub.1])[a.sub.1] is bounded.

Using (3) we may write, for example:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In order to prove our theorem it is sufficient, by Minkowski's inequality, to show that

[[infinity].summation over (n=2)] [n.sup.[delta]k+k-1] [[absolute value of [T.sub.nr]].sup.k] < [infinity], for r = 1, 2, 3, 4.

Using Holder's inequality, (iii), (v), and (vii)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Using Holder's inequality, (iii), (iv), (vi), and (viii)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Using Holder's inequality, (iii), (iv), (vi), and (ix)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Finally, using (iii) and (v)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Setting [delta] = 0 in the theorem yields the following corollary:

Corollary 1 ([2]). Let A be a triangle satisfying conditions (i)-(iv) of Theorem 1 and let [t.sup.1.sub.n] denote the nth (C, 1) mean of {[na.sub.n]}. If

(v) [[infinity].summation over (v=1)] [a.sub.vv][[absolute value of [[lambda].sub.v]].sup.k] [[absolute value of [t.sup.1.sub.v]].sup.k] = O(1),

(vi) [[infinity].summation over (v=1)] [absolute value of [a.sub.vv]].sup.1-k][[absolute value of [DELTA][[lambda].sub.v]].sup.k] [[absolute value of [t.sup.1.sub.v]].sup.k] = O(1),

(vii) [[infinity].summation over (v=1)] [a.sub.vv][[absolute value of [[lambda].sub.v+1]].sup.k] [[absolute value of [t.sup.1.sub.v]].sup.k] = O(1),

then the series [summation] [a.sub.n][[lambda].sub.n] is summable [[absolute value of A].sup.k], k [member of] N.

Remark. I must note that in the theorem of Savas [2], the following condition should be added.

(vii) [[infinity].summation over (v=1)] [a.sub.vv][[absolute value of [[lambda].sub.v+1]].sup.k] [[absolute value of [t.sup.1.sub.v]].sup.k] = O(1).

3. CONCLUSION

Let [summation][a.sub.v] denote a series with partial sums [s.sub.n]. For an infinite matrix T, the nth term of the T- transform of {[s.sub.n]} is denoted by

[t.sub.n] = [[infinity].summation over (v=0)] [t.sub.nv][s.sub.v].

Let [[sigma].sup.[alpha].sub.n] denote the nth terms of the transform of a Cesaro matrix (C, [alpha]) of a sequence {[s.sub.n]}. In 1957 Fleet [1] gave the following definition. A series [summation][a.sub.n], with partial sums [s.sub.n], is said to be absolutely (C, [alpha]) summable of order k[greater than or equal to] 1, written [summation][a.sub.n] is summable [[absolute value of C, [alpha]].sub.k], if

[[infinity].summation over (n=1)][n.sup.k-1][[absolute value of [[sigma].sup.[alpha].sub.n-1] - [[sigma].sup.[alpha].sub.n]].sup.k] < [infinity]. (4)

Recently, Savas [2] obtained an absolute summability factor theorem for lower triangular matrices. A summability factor theorem for summability [[absolute value of A, [delta]].sub.k] has not been studied so far. The present paper fills up a gap in the existing literature.

Olgu A kolmnurkne maatriks k [greater than or equal to] 1 ja [delta] [greater than or equal to] 0. Artiklis on defineeritud maatriksmenetlusega A k-jarku uldistatud absoluutse summeeruvuse ehk [[absolute value of A, [delta]].sub.k]-summeeruvuse moiste jaleitud piisavad tingimused selleks, et rida [summation][[lambda].sub.n][a.sub.n] oleks [[absolute value of A, [delta]].sub.k]-summeeruv, st et arvud [[lambda].sub.n] oleksid menetluse A k-jarku uldistatud absoluutse summeeruvuse teguriteks. Saadud tulemus uldistab autori varasemat tulemust

doi: 10.3176/proc.2011.2.06

ACKNOWLEDGEMENTS

I wish to thank the referees for their careful reading of the manuscript and for their helpful suggestions.

REFERENCES

[1.] Fleet, T. M. On an extension of absolute summability and some theorems of Littlewood and Paley. Proc. London Math. Soc., 1957, 3(7), 113-141.

[2.] Savas, E. A study on absolute summability factors for a triangular matrix. Math. Ineq. Appl., 2009, 12(19), 141- 146.

Ekrem Savas

Department of Mathematics, Istanbul Ticaret University, Uskudar-Istanbul, Turkey; ekremsavas@yahoo.com; esavas@iticu.edu.tr

Received 3 August 2009, revised 26 March 2010, accepted 5 April 2010

Savas [2] obtained sufficient conditions for [summation][a.sub.n][[lambda].sub.n] to be [[absolute value of A].sub.k]- summable, k [member of] N. In this paper, a theorem on [[absolute value of A, [delta]]-summability methods is proved. This theorem includes a known result.

Let A be a lower triangular matrix, {[s.sub.n]} a sequence. Then

[A.sub.n] : = [n.summation over (v=0)] [a.sub.nv][s.sub.v].

A series [summation][a.sub.n] is said to be summable [[absolute value of A].sub.k], k [greater than or equal to] 1 if

[[infinity].summation over (n=1)][n.sub.k-1][[absolute value of [A.sub.n] - [A.sub.n-1]].sup.k] < [infinity], (1)

and it is said to be summable [[absolute value of A, [delta]].sub.k] k [greater than or equal to] 1 and [delta] [greater than or equal to] 0 if (see, [1])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

We may associate with A two lower triangular matrices [bar.A] and [??] defined as follows:

[[bar.a].sub.nv] = [n.summation over (r=v)] [a.sub.nr], n, v = 0, 1, 2, ...,

and

[[??].sub.nv] = [[bar.n].sub.nv] - [[bar.a].sub.n-1,v], n = 1, 2, 3,....

Also we shall define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

Given any sequence {[x.sub.n]}, the notation [x.sub.n] [??] O(1) means [x.sub.n] = O(1) and 1/[x.sub.n] = O(1). For any matrix entry [a.sub.nv], [[DELTA].sub.v][a.sub.nv] := [a.sub.nv] - [a.sub.nv+1].

2. MAIN RESULT

Theorem 1. Let A be a lower triangular matrix with nonnegative entries such that

(i) [[bar.a].sub.n0] = 1, n = 0, 1, ...,

(ii) [a.sub.n-1,v] [greater than or equal to] [a.sub.nv] for n [greater than or equal to] v + 1,

(iii) [na.sub.nn] [??] O(1), and

(iv) [[summation].sup.n=1.sub.v=1] [a.sub.vv][absolute value of [[??].sub.n,v+1]] = O([a.sub.nn]).

(v) [m+1.summation over (n=v+1)] [n.sup.[delta]k][absolute value of [[DELTA].sub.v][[??].sub.nv]] = O([v.sup.[delta]k] [a.sub.vv]) and

(vi) [m+1.summation over (n=v+1)] [n.sup.[delta]k][[??].sub.nv+1]] = O([v.sup.[delta]k]).

Let [t.sup.1.sub.n] denote the nth (C, 1) mean of {[na.sub.n]}. If

(vii) [[infinity].summation over (v=1)] [v.sup.[delta]k][a.sub.vv][[absolute value of [[lambda].sub.v]].sup.k] [[absolute value of [t.sup.1.sub.v]].sup.k] = O(1),

(viii) [[infinity].summation over (v=1)] [v.sup.[delta]k][absolute value of [a.sub.vv]].sup.1-k][[absolute value of [DELTA][[lambda].sub.v]].sup.k] [[absolute value of [t.sup.1.sub.v]].sup.k] = O(1),

(ix) [[infinity].summation over (v=1)] [v.sup.[delta]k][a.sub.vv][[absolute value of [[lambda].sub.v+1]].sup.k] [[absolute value of [t.sup.1.sub.v]].sup.k] = O(1),

then the series [summation][a.sub.n][[lambda].sub.n] is summable [[absolute value of A, [delta]].sub.k], k [greater than or equal to] 1, 0 [less than or equal to] [delta] < 1/k.

Proof. From (i) it follows that [[??].sub.n,0] = 0. Also ([[??].sub.n1] [[lambda].sub.1])[a.sub.1] is bounded.

Using (3) we may write, for example:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In order to prove our theorem it is sufficient, by Minkowski's inequality, to show that

[[infinity].summation over (n=2)] [n.sup.[delta]k+k-1] [[absolute value of [T.sub.nr]].sup.k] < [infinity], for r = 1, 2, 3, 4.

Using Holder's inequality, (iii), (v), and (vii)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Using Holder's inequality, (iii), (iv), (vi), and (viii)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Using Holder's inequality, (iii), (iv), (vi), and (ix)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Finally, using (iii) and (v)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Setting [delta] = 0 in the theorem yields the following corollary:

Corollary 1 ([2]). Let A be a triangle satisfying conditions (i)-(iv) of Theorem 1 and let [t.sup.1.sub.n] denote the nth (C, 1) mean of {[na.sub.n]}. If

(v) [[infinity].summation over (v=1)] [a.sub.vv][[absolute value of [[lambda].sub.v]].sup.k] [[absolute value of [t.sup.1.sub.v]].sup.k] = O(1),

(vi) [[infinity].summation over (v=1)] [absolute value of [a.sub.vv]].sup.1-k][[absolute value of [DELTA][[lambda].sub.v]].sup.k] [[absolute value of [t.sup.1.sub.v]].sup.k] = O(1),

(vii) [[infinity].summation over (v=1)] [a.sub.vv][[absolute value of [[lambda].sub.v+1]].sup.k] [[absolute value of [t.sup.1.sub.v]].sup.k] = O(1),

then the series [summation] [a.sub.n][[lambda].sub.n] is summable [[absolute value of A].sup.k], k [member of] N.

Remark. I must note that in the theorem of Savas [2], the following condition should be added.

(vii) [[infinity].summation over (v=1)] [a.sub.vv][[absolute value of [[lambda].sub.v+1]].sup.k] [[absolute value of [t.sup.1.sub.v]].sup.k] = O(1).

3. CONCLUSION

Let [summation][a.sub.v] denote a series with partial sums [s.sub.n]. For an infinite matrix T, the nth term of the T- transform of {[s.sub.n]} is denoted by

[t.sub.n] = [[infinity].summation over (v=0)] [t.sub.nv][s.sub.v].

Let [[sigma].sup.[alpha].sub.n] denote the nth terms of the transform of a Cesaro matrix (C, [alpha]) of a sequence {[s.sub.n]}. In 1957 Fleet [1] gave the following definition. A series [summation][a.sub.n], with partial sums [s.sub.n], is said to be absolutely (C, [alpha]) summable of order k[greater than or equal to] 1, written [summation][a.sub.n] is summable [[absolute value of C, [alpha]].sub.k], if

[[infinity].summation over (n=1)][n.sup.k-1][[absolute value of [[sigma].sup.[alpha].sub.n-1] - [[sigma].sup.[alpha].sub.n]].sup.k] < [infinity]. (4)

Recently, Savas [2] obtained an absolute summability factor theorem for lower triangular matrices. A summability factor theorem for summability [[absolute value of A, [delta]].sub.k] has not been studied so far. The present paper fills up a gap in the existing literature.

Olgu A kolmnurkne maatriks k [greater than or equal to] 1 ja [delta] [greater than or equal to] 0. Artiklis on defineeritud maatriksmenetlusega A k-jarku uldistatud absoluutse summeeruvuse ehk [[absolute value of A, [delta]].sub.k]-summeeruvuse moiste jaleitud piisavad tingimused selleks, et rida [summation][[lambda].sub.n][a.sub.n] oleks [[absolute value of A, [delta]].sub.k]-summeeruv, st et arvud [[lambda].sub.n] oleksid menetluse A k-jarku uldistatud absoluutse summeeruvuse teguriteks. Saadud tulemus uldistab autori varasemat tulemust

doi: 10.3176/proc.2011.2.06

ACKNOWLEDGEMENTS

I wish to thank the referees for their careful reading of the manuscript and for their helpful suggestions.

REFERENCES

[1.] Fleet, T. M. On an extension of absolute summability and some theorems of Littlewood and Paley. Proc. London Math. Soc., 1957, 3(7), 113-141.

[2.] Savas, E. A study on absolute summability factors for a triangular matrix. Math. Ineq. Appl., 2009, 12(19), 141- 146.

Ekrem Savas

Department of Mathematics, Istanbul Ticaret University, Uskudar-Istanbul, Turkey; ekremsavas@yahoo.com; esavas@iticu.edu.tr

Received 3 August 2009, revised 26 March 2010, accepted 5 April 2010

Gale Copyright:

Copyright 2011 Gale, Cengage Learning. All rights
reserved.