The Cesáro Core of Double Sequences

Abstract

We have characterized a new type of core for double sequences, -core, and determined the necessary and sufficient conditions on a four-dimensional matrix to yield -core-core for all .

1. Introduction

A double sequence is said to be convergent in the Pringsheim sense or -convergent if for every there exists an such that whenever , [1]. In this case, we write . By , we mean the space of all -convergent sequences.

A double sequence is bounded if By , we denote the space of all bounded double sequences.

Note that, in contrast to the case for single sequences, a convergent double sequence need not be bounded. So, we denote by the space of double sequences which are bounded and convergent.

A double sequence is said to converge regularly if it converges in Pringsheim's sense and, in addition, the following finite limits exist: Let be a four-dimensional infinite matrix of real numbers for all . The sums are called the -transforms of the double sequence . We say that a sequence is -summable to the limit if the -transform of exists for all and is convergent to in the Pringsheim sense, that is,

We say that a matrix is bounded-regular if every bounded-convergent sequence is -summable to the same limit and the -transforms are also bounded. The necessary and sufficient conditions for to be bounded-regular or RH-regular (cf., Robison [2]) are A double sequence is said to be almost convergent (see [3]) to a number if Let be a one-to-one mapping from into itself. The almost convergence of double sequences has been generalized to the -convergence in [4] as follows: where . In this case, we write . By , we denote the set of all -convergent and bounded double sequences. One can see that in contrast to the case for single sequences, a convergent double sequence need not be -convergent. But every bounded convergent double sequence is -convergent. So, . In the case , -convergence of double sequences reduces to the almost convergence. A matrix is said to be -regular if for with , and we denote this by , (see [5, 6]). Mursaleen and Mohiuddine defined and characterized -conservative and -coercive matrices for double sequences [6].

A double sequence of real numbers is said to be Cesáro convergent (or -convergent) to a number L if and only if , where We shall denote by the space of Cesáro convergent (-convergent) double sequences.

A matrix is said to be -multiplicative if for with , and in this case we write . Note that if , then -multiplicative matrices are said to be -regular matrices.

Recall that the Knopp core (or K-core) of a real number single sequence is defined by the closed interval , where and . The well-known Knopp core theorem states (cf., Maddox [7] and Knopp [8]) that in order that for every bounded real sequence , it is necessary and sufficient that should be regular and . Patterson [9] extended this idea for double sequences by defining the Pringsheim core (or P-core) of a real bounded double sequence as the closed interval . Some inequalities related to the these concepts have been studied in [5, 9, 10]. Let

Then, MR- (Moricz-Rhoades) and -core of a double sequence have been introduced by the closed intervals and , and also the inequalities have been studies in [3–5, 11].

In this paper, we introduce the concept of -multiplicative matrices and determine the necessary and sufficient conditions for a matrix to belong to the class . Further we investigate the necessary and sufficient conditions for the inequality for all .

2. Main Results

Let us write Then, we will define the -core of a realvalued bounded double sequence by the closed interval . Since every bounded convergent double sequence is Cesáro convergent, we have , and hence it follows that -core-core for a bounded double sequence .

Lemma 2.1 .. A matrix is -multiplicative if and only if where the means and

Proof. Sufficiency. Suppose that the conditions (2.2)-(2.6) hold and with , say. So that for every there exists such that whenever .
Then, we can write Therefore, Letting and using the conditions (2.2)–(2.6), we get Since is arbitrary, . Hence , that is, is -multiplicative.

Necessity. Suppose that A is -multiplicative. Then, by the definition, the A-transform of exists and for each . Therefore, is also bounded. Then, we can write for each . Now, let us define a sequence by  . Then, the necessity of (10) follows by considering the sequence in (2.11).
Also, by the assumption, we have Now let us define the sequence as follows: and write , . Then, the necessity of (2.2), (2.4), and (2.5) follows from , and , respectively.

Note that when , the above theorem gives the characterization of . Now, we are ready to construct our main theorem.

Theorem 2.2. For every bounded double sequence , or if and only if is -multiplicative and

Proof. Necessity. Let (2.15) hold and for all . So, since , then, we get That is, where By choosing , we get from (2.17) that This means that is -multiplicative.
By Lemma  3.1 of Patterson [9], there exists a with such that If we choose , it follows Since , we have from (2.15) that This gives the necessity of (2.16).

Sufficiency. Suppose that is -regular and (2.16) holds. Let be an arbitrary bounded sequence. Then, there exist such that for all . Now, we can write the following inequality: Using the condition of -multiplicative and condition (2.16), we get This completes the proof of the theorem.

Theorem 2.3. For , if , then .

Proof. Since , we have and . Using definition of , we take . Since is sublinear, Therefore, . Since , this implies that .By an argument similar as above, we can show that . This completes the proof.

Acknowledgment

The authors would like to state their deep thanks to the referees for their valuable suggestions improving the paper.

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Copyright

Copyright © 2011 Kuddusi Kayaduman and Celal Çakan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.