A new note on generalized absolute matrix summability
© Özarslan and Ari; licensee Springer 2012
Received: 3 June 2012
Accepted: 13 July 2012
Published: 27 July 2012
This paper gives necessary and sufficient conditions in order that a series should be summable , , whenever is summable . Some new results have also been obtained.
MSC:40D25, 40F05, 40G99.
Keywordssummability factors absolute matrix summability infinite series
If we take , then summability is the same as summability.
Before stating the main theorem we must first introduce some further notations.
If A is a normal matrix, then will denote the inverse of A. Clearly if A is normal, then is normal and has two-sided inverse , which is also normal (see ).
Sarıgöl  has proved the following theorem for summability method.
and we regarded that the above series converges for each v and Δ is the forward difference operator.
It may be remarked that the above theorem has been proved by Orhan and Sarıgöl .
for the cases, wheredenotes the set of all matrices A which mapinto.
2 Main theorem
The aim of this paper is to generalize Theorem A for the and summabilities. Therefore we shall prove the following theorem.
are sufficient for the consequent to hold.
It should be noted that if we take and , then we get Theorem A. Also if we take , then we get Theorem B.
which is condition (21).
This completes the proof of the Theorem. □
- Bor H: On the relative strength of two absolute summability methods. Proc. Am. Math. Soc. 1991, 113: 1009–1012.MathSciNetView ArticleMATHGoogle Scholar
- Cooke RG: Infinite Matrices and Sequence Spaces. Macmillan & Co., London; 1950.MATHGoogle Scholar
- Hardy GH: Divergent Series. Oxford University Press, Oxford; 1949.MATHGoogle Scholar
- Maddox IJ: Elements of Functional Analysis. Cambridge University Press, Cambridge; 1970.MATHGoogle Scholar
- Orhan C, Sarıgöl MA: On absolute weighted mean summability. Rocky Mt. J. Math. 1993, 23(3):1091–1098. 10.1216/rmjm/1181072543View ArticleMathSciNetMATHGoogle Scholar
- Sarıgöl MA: On the absolute Riesz summability factors of infinite series. Indian J. Pure Appl. Math. 1992, 23(12):881–886.MathSciNetMATHGoogle Scholar
- Tanovic̆-Miller N: On strong summability. Glas. Mat. 1979, 34: 87–97.MathSciNetGoogle Scholar
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