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A new note on generalized absolute matrix summability
Journal of Inequalities and Applications volume 2012, Article number: 166 (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.
Let be a given infinite series with the partial sums . Let be a sequence of positive numbers such that
The sequence-to-sequence transformation
defines the sequence of the Riesz means of the sequence generated by the sequence of coefficients (see ). The series is said to be summable , if (see )
Let be a normal matrix, i.e., a lower triangular matrix of nonzero diagonal entries. Then A defines the sequence-to-sequence transformation, mapping the sequence to , where
The series is said to be summable , if (see )
If we take , then summability is the same as summability.
Before stating the main theorem we must first introduce some further notations.
Given a normal matrix , we associate two lover semimatrices and as follows:
It may be noted that and are the well-known matrices of series-to-sequence and series-to-series transformations, respectively. Then, we have
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.
Theorem A Suppose thatandare positive sequences withandas. Thenis summable, wheneveris summable, if and only if
Theorem B Thesummability implies the, , summability if and only if the following conditions hold:
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 .
if and only if
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.
Theorem Let, andbe two positive normal matrices such that
Then, in order thatis summablewheneveris summable, it is necessary that
Also (19)-(21) and
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.
Proof of the Theorem Necessity. Let and denote A-transform and B-transform of the series and , respectively. Then, by (8) and (9), we have
For , we define
Then it is routine to verify that these are BK-spaces, if normed by
respectively. Since is summable implies is summable , by the hypothesis of the theorem,
Now consider the inclusion map defined by . This is continuous, which is immediate as A and B are BK-spaces. Thus there exists a constant M such that
By applying (25) to ( is the v th coordinate vector), we have
So (26) and (27) give us
Hence it follows from (28) that
Using (17), we can find
The above inequality will be true if and only if each term on the left-hand side is . Taking the first term,
which verifies that (19) is necessary. Using the second term, we have
which is condition (20). Now, if we apply (22) to , we have
Hence it follows from (28) that
Using (18) we can find
which is condition (21).
Sufficiency. We use the notations of necessity. Then
In this case
On the other hand, since
by (22), we have
By considering the equality
where is the Kronecker delta, we have that
to complete the proof of Theorem, it is sufficient to show that
by Lemma. But it follows from conditions (20), (21) and (23) that
by Lemma. But it follows from conditions (21) and (24) that
Therefore, we have
This completes the proof of the Theorem. □
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Özarslan, H., Ari, T. A new note on generalized absolute matrix summability. J Inequal Appl 2012, 166 (2012). https://doi.org/10.1186/1029-242X-2012-166
- summability factors
- absolute matrix summability
- infinite series