Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 682823, 5 pages

http://dx.doi.org/10.1155/2013/682823

Research Article

Statistical Convergence of Double Sequences of Order

R. Çolak and Y. Altin

Department of Mathematics, Fırat University, 23119 Elazığ, Turkey

Received 14 May 2013; Accepted 10 July 2013

Academic Editor: Dachun Yang

Copyright © 2013 R. Çolak and Y. Altin. 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.

Abstract.

We intend to make a new approach and introduce the concepts of statistical convergence of order and strongly -Cesàro summability of order for double sequences of complex or real numbers. Also, some relations between the statistical convergence of order and strong -Cesàro summability of order are given.

1. Introduction

The concept of statistical convergence was introduced by Steinhaus [1] and Fast [2] and later investigated by Schoenberg [3] independently for real and complex sequences. Over the years and under different names, statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory, number theory, measure theory, trigonometric series, turnpike theory, and Banach spaces. Later on, it was further investigated from the sequence space point of view and linked with summability theory by Fridy [4], Connor [5], Maddox [6], Rath and Tripathy [7], Šalát [8], Tripathy et al. [9–12], and many others.

The idea of statistical convergence was later extended to double sequences by Mursaleen and Edely [13], Karakaya et al. [14], Móricz [15], and Tripathy et al. [16–18]. More recent developments on double sequences can be found in [19–22].

In the present paper, we introduce and examine the concepts of statistically convergence of double sequences of order and strong -Cesàro summability of order of double sequences of complex or real numbers, where denotes the pair , and , are any real numbers such that . The order of statistical convergence of a single sequence of numbers was given in [23], and after that, statistical convergence of order and strong -Cesàro summability of order were studied by Çolak in [24, 25].

In Section 2, we give a brief information about statistical convergence and strong -Cesàro summability, and we define the concepts of statistical convergence of order and strong -Cesàro summability of order and give some results. In Section 3, we give the main results and establish some inclusion relations between and .

2. Definitions and Preliminaries

In this section, mainly we introduce and examine the concepts of the -double density of a subset of , statistical convergence of order , and strong -Cesàro summability of order of the double sequences of complex or real numbers for .

denotes the space of all double sequences. Let , and be the linear spaces of bounded, convergent, and null sequences with complex terms, respectively, normed by , where the set of positive integers.

By the convergence of a double sequence, we mean the convergence in Pringsheim’s sense [26]. A double sequence is said to be convergent in the Pringsheim sense if for every there exists such that whenever ; is called the Pringsheim limit of .

A double sequence is bounded if there exists a positive number such that for all and , that is, if

Note that in contrast to the case for single sequences, a convergent double sequence need not be bounded.

Let and let be the number of in such that and . Then, the lower asymptotic density of a set is defined by

In case the sequence has a limit in Pringsheim’s sense, then we say that has a double natural density and is defined by

Throughout the paper, we take , otherwise indicated, and for the sake of brevity, we write instead of and instead of . Also, we define

   and   and   and     in case   in case   in case and ,

and furthermore, we write to denote and to denote in the following.

Now we define the -double density of the set as

Remark 1. Note that for any set , may be greater than , even equal to , but . Also, holds but does not hold in general.

Definition 2. Let and be given. The sequence is said to be statistically convergent of order if there is a complex number such that for every ,

which is the case when we say that is statistically convergent of order to . In this case, we write and we denote the set of all statistically convergent double sequences of order by .

In case , the statistical convergence of order reduces to the statistical convergence of double sequences [13]. If is statistically convergent of order to the number , then is determined uniquely. The statistical convergence of order is well defined for , but it is not well defined for . For this, let be defined as follows:

Then,

for that is and , so that statistically converges of order both to and , that is, and . However, this is impossible.

Theorem 3. Let and , be sequences of complex numbers. Then,

(i)If and , then (ii)If and , then .

Proof. (i) is clear in case . Suppose that ; then the proof of (i) follows from

and that of (ii) follows from the following inequality:

It is easy to see that every convergent double sequence is statistically convergent of order to the same number, that is for each that is for each pair of such that . However, the converse does not hold. For example, the sequence defined by

is statistically convergent of order with for that is , but it is not convergent.

Definition 4. Let be given. A sequence is said to be Cesàro summable of order if there is a complex number such that

which is the case when we say that is Cesàro summable of order to . For , the Cesàro summability of order reduces to the Cesàro summability that is given in [15]. The set of all Cesàro summable double sequences of order will be denoted by . The set of all Cesàro summable double sequences will be denoted by .

Definition 5. Let be given, and let be a positive real number. Then, a sequence is said to be strongly -Cesàro summable of order if there is a complex number such that

which is the case when we say that is strongly -Cesàro summable of order to . For , the strong -Cesàro summability of order reduces to the strong -Cesàro summability that is given in [13]. The set of all strongly -Cesàro summable sequences of order will be denoted by .

3. Main Results

In this section, we give the main results of the paper. In Theorem 6, we give the relationship between the statistical double convergence of order and the statistical double convergence of order for and so that the relationship between the statistical double convergence of order and the statistical convergence. In Corollary 10, we give the relationship between the strong -Cesàro summability of order and the strong -Cesàro summability of order . In Theorem 12, we give the relationship between the strong -Cesàro summability of order and the statistical double convergence of order .

Theorem 6. Let be given such that . Then, and the inclusion is strict for some and such that .

Proof. Let be given. If and so that and , then

for every and this gives that . To show that the inclusion is strict, consider the sequence defined by

Then, ; that is, for (i.e., for and but for (i.e., for and ).

If we take in Theorem 6, then we obtain the following result.

Corollary 7. If a double sequence is statistically convergent of order to , for some such that , then it is statistically convergent to ; that is , and the inclusion is strict.

From Theorem 6, we have the following results and the proof is easy.

Corollary 8. Let be given. Then,

(i) if and only if ;(ii) if and only if .

The proof of the following theorem is straight forward from the definition, so omitted.

Theorem 9. Let be given such that , and let be a positive real number. Then, and the inclusion is strict for some and such that .

Proof. Let . Then, given and such that and a positive real number , we may write

and this gives that .

To show that the inclusion is strict, consider the sequence defined in (14). It is easy to see that

Since as , , then for (i.e., for and ), but since

and as , , then for (i.e., for and ). This completes the proof.

The following result is a consequence of Theorem 9.

Corollary 10. Let be given such that , and let be a positive real number. Then,

(i) if and only if ;(ii)  for each such that and .

The following result is a simple consequence of Hölder’s inequality which is an extension of a result of Maddox [27].

Theorem 11. Let , and let . Then, .

Taking in Theorem 11, we obtain a result of Maddox [27]: if , then .

Theorem 12. Let and be given such that , and let , where . If a sequence is strongly -Cesàro summable of order to , then it is statistically convergent of order to .

Proof. For any sequences and , we have

so that since ,

From this, it follows that if is strongly -Cesàro summable of order to , then it is statistically convergent of order to .

If we take in Theorem 12, we obtain the following result.

Corollary 13. Let be given, and let . If a double sequence is strongly -Cesàro summable of order to , then it is statistically convergent of order to .

Remark 14. Note that the converse of Theorem 12 does not hold in general. We see that a bounded and statistically convergent double sequence of order need not be strongly -Cesàro summable of order in general for .

The sequence defined by

is an example for this case. It is clear that and for each . First, recall that the inequality

holds for every positive integer and . Define , and take . Since

we have

so that for if . Therefore, for if .

Corollary 15. Let , and let be a positive real number. Then, . The inclusion is strict if .

Proof. From Corollaries 13 and 7, we have . To show that the inclusion is strict, consider the sequence defined in (10). Then, clearly ; that is, but for and . Indeed it is easy to see that

Since as and as , then if and . Consequently, for and . This completes the proof.

4. Conclusion

The concepts of statistical convergence and strong Cesàro summability of double sequences of complex or real numbers have been studied by various mathematicians. In this paper, we introduced the concepts of statistical convergence of order and strong -Cesàro summability of order for double sequences, where . These concepts are much more general than the concepts of statistical convergence and strong -Cesàro summability of double sequences that include these concepts in the special case .

Note that the converse of Theorem 12 does not hold in general; that is, a statistically convergent sequence of order (even a bounded and statistically convergent sequence of order ) need not be strongly -Cesàro summable of order in general.

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