Almost Conservative Four-Dimensional Matrices through de la Vallée-Poussin Mean

Abstract

The purpose of this paper is to generalize the concept of almost convergence for double sequence through the notion of de la Vallée-Poussin mean for double sequences. We also define and characterize the generalized regularly almost conservative and almost coercive four-dimensional matrices. Further, we characterize the infinite matrices which transform the sequence belonging to the space of absolutely convergent double series into the space of generalized almost convergence.

1. Introduction and Preliminaries

Let be the Banach space of real bounded sequences with the usual norm . There exist continuous linear functionals on called Banach limits [1]. It is well known that any Banach limit of lies between and . The idea of almost convergence of Lorentz [2] is narrowly connected with the limits of S. Banach; that is, a sequence is almost convergent to if all of its Banach limits are equal. As an application of almost convergence, Mohiuddine [3] obtained some approximation theorems for sequence of positive linear operator through this notion. For double sequence, the notion of almost convergence was first introduced by Móricz and Rhoades [4]. The authors of [5] introduced the notion of Banach limit for double sequence and characterized the spaces of almost and strong almost convergence for double sequences through some sublinear functionals. For more details on these concepts, one can refer to [6–12].

We say that a double sequence of real or complex numbers is bounded if denoted by , the space of all bounded sequence .

A double sequence of reals is called convergent to some number in Pringsheim’s sense (briefly, -convergent to ) [13] if for every there exists such that whenever , where .

If a double sequence in and is also -convergent to , then we say that it is boundedly -convergent to (briefly, -convergent to ).

A double sequence is said to converge regularly to (briefly, -convergent to ) if is converges in Pringsheim’s sense, and the limits and exist. Note that in this case the limits and exist and are equal to the -limit of .

Throughout this paper, by , , and , we denote the space of all -convergent, -convergent, and -convergent double sequences, respectively. Also, the linear space of all continuous linear functionals on is denoted by .

Let be a four-dimensional infinite matrix of real numbers for all , and a space of double sequences. Let be a double sequences space, converging with respect to a convergence rule . Define Then, we say that a four-dimensional matrix maps the space into the space if and is denoted by .

Móricz and Rhoades [4] extended the notion of almost convergence from single to double sequence and characterized some matrix classes involving this concept. A double sequence of real numbers is said to be almost convergent to a number if

For more details on double sequences and 4-dimensional matrices, one can refer to [14–20].

Using the notion of almost convergence for single sequence, King [21] introduced a slightly more general class of matrices than the conservative and regular matrices, that is, almost conservative and almost regular matrices, and presented its characterization. In [22], Schaefer presented some interesting characterization for almost convergence. The Steinhaus-type theorem for the concepts of almost regular and almost coercive matrices was proved by Başar and Solak [23]. In this paper, we generalize the concept of almost convergence for double sequences with the help of double generalized de la Vallée-Poussin mean and called it almost convergence. Using this concept, we define the notions of regularly almost conservative and almost coercive four-dimensional matrices and obtain their necessity and sufficient conditions. Further, we introduce the space of all absolutely convergent double series and characterize the matrix class , where denotes the space of almost convergence for double sequences.

2. Main Results

Definition 1. Let and be two nondecreasing sequences of positive reals with each tending to such that , , , , and is called the double generalized de la Vallée-Poussin mean, where and . We denote the set of all and type sequences by using the symbol .

Definition 2. A double sequence of reals is said to be almost convergent (briefly, -convergent) to some number if , where denoted by , the space of all almost convergent sequences . Note that .

Remark 3. If we take and , then the notion of almost convergence reduced to almost convergence due to Móricz and Rhoades [4].

Definition 4. A four-dimensional matrix is said to be regularly almost conservative if it maps every -convergent double sequence into -convergent double sequence; that is, . In addition, if , then is regularly almost regular.

Definition 5. A matrix is said to be almost coercive if it maps every -convergent double sequence into -convergent double sequence, briefly, a matrix in .

Theorem 6. A matrix is regularly almost conservative if and only if (CR₁),(CR₂), for each , (uniformly in , ),(CR₃), for each (uniformly in , ),(CR₄), for each (uniformly in , ),(CR₅), (uniformly in , ), where In this case, the -limit of is where .

Proof. Necessity. Suppose that is regularly almost conservative matrix. Fix , , the set of integers. Let where It is clear that Hence for , . Since is regularly almost conservative, we have uniformly in , . It follows that is bounded for and fixed , . Hence, is bounded by the uniform boundedness principle.
For each , , define the sequence by Then, a double sequence , , and Hence Therefore so that condition () follows.
The sequences , , , and are defined by Since , , , , the -limit of , , , and must exist, uniformly in , . Hence, the conditions ()–() must hold, respectively.
Sufficiency. Suppose that the conditions ()–() hold and a double sequence . Fix , . Then Therefore, by (), we have , where is a constant independent of , . Hence and the sequence is bounded for each . It follow from the conditions (), (), (), and () that the -limit of , , , and exist for all , , , and . Since {, , and } is a fundamental set in (see [24]), it follows that exists and . Therefore, has the form But , , , and by the conditions ()–(), respectively. Hence exists for each and , with Since for each , , , and , it has the form It is easy to see from (21) and (22) that the convergence of to is uniform in , , since , , , and uniformly in , . Therefore, is regularly almost conservative.

Let us recall the following lemma, which is proved by Mursaleen and Mohiuddine [25].

Lemma 7. Let , , , be a sequence of infinite matrices such that (i) for all , ; and(ii)for each , uniformly in , .Then if and only if

Theorem 8. A matrix is almost coercive if and only if (AC₁),(AC₂), for each , (uniformly in , ),(AC₃) , uniformly in , .In this case, the -limit of is for every .

Proof. Sufficiency. Assume that conditions ()–() hold. For any positive integers , This shows that converges and that is defined for every double sequence .
Let be any arbitrary bounded double sequence. For every positive integers , Letting , and using condition (), we get Hence, with .
Necessity. Let be almost coercive matrix. This implies that a four-dimensional matrix is almost conservative; then we have conditions () and () from Theorem 6. Now we have to show that () holds.
Suppose that, for some , , we have Since is finite, therefore is also finite. We observe that since and is almost coercive, the matrix , where is also almost coercive matrix. By an argument similar to that of Theorem  2.1 in [26] for single sequences, one can find for which . This contradiction implies the necessity of ().
Now, we use Lemma 7 to show that this convergence is uniform in , . Let and let be the matrix . It is easy to see that for every , ; and from condition () For any , and the limit exists uniformly in , , since . Moreover, this limit is zero since Hence This shows that matrix satisfies condition ().

In the following theorem, we characterize the four-dimensional matrices of type , where the space of all absolutely convergent double series.

Theorem 9. A matrix if and only if it satisfies the following conditions:(i),and the condition () of Theorem 6 holds.

Proof. Sufficiency. Suppose that conditions (i) and () hold. For any double sequence , we see that uniformly in , and it also converges absolutely. Furthermore, converges absolutely for each , , , and . Given , there exist and such that By the condition (), we can find , such that for all and , uniformly in , . Now, by using the conditions (37), (38), and (), we get for all , and uniformly in , . Hence (37) holds.
Necessity. Suppose that . The condition () follows from the fact that , where with for all , . To verify the condition (i), we define a continuous linear functional on by Now and hence For any fixed , we define a double sequence by Then , and so that It follows from (42) and (45) that Since , we have Hence, by the uniform boundedness principle, we obtain

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. (130-100-D1434). The authors, therefore, acknowledge with thanks DSR technical and financial support.