Abstract
In this paper, we introduce a new type of statistical convergence method for double sequences by using the -method of summability which was defined by Natarajan. We also obtain some inclusion relations between statistical convergence and -statistical convergence for double sequences.
1. Introduction
The subject of statistical convergence has been studied by many researchers since the emergence of the idea of statistical convergence in 1935. Statistical convergence was introduced by Fast [2] and Steinhaus [3] independently in the same year 1951 as a generalization of ordinary convergence and was later reintroduced by Schoenberg [4]. Quite a few researchers have generalized or extended this concept and applied different fields of mathematics such as Erdös and Tenenbaum [5], Miller [6], Zygmund [7], Freedmanet al. [8], Connor [9], Salat [10], Duman and Orhan [11], Et et al. [12], Çakallı [13, 14], Çakallı and Savaş [15], Edely et al. [16], Mursaleen et al. [17, 18], Natarajan [19], Tok and Başarır [20], Aral and Küçükaslan [21–23], and Taylan [24].
Let be a double sequence. Then, is said to be convergent to in the Pringsheim sense if for every , there exists such that , whenever . In this case, we write [25].
Also, a double sequence is said to be bounded if there exists a positive number such that for all . By , we denote the set of all bounded double sequences.
Let and . Then, the double natural density of is given byif the limit exists. A double sequence is said to be statistically convergent to provided that, for every , the sethas double natural density zero [26]. In this case, we write . By , we denote the set of all statistically convergent double sequences. Later, a lot of works have been done on the statistical convergence of double sequences (see [27–33]).
The following definition which is required for our study is given by Natarajan.
Definition 1 (see [1]). Let be a double sequence such that . Then, the -method is defined by the 4-dimensional infinite matrix , whereIt is well known from Theorem 3.4 of [1] that the -method is regular if and only if .
The purpose of this paper is to give a new statistical convergence definition using the above definition for double sequences and some relations between statistical convergence and -statistical convergence. In addition, we have used it to prove a Korovkin-type approximation theorem as an application of our method.
We acknowledge that the definition of -statistical convergence for double sequences was presented at the International Conference on Multidisciplinary, Engineering, Science, Education and Technology in 2017 [34].
Let be a given real-valued sequence. A sequence of -mean of is defined byfor all .
Definition 2. The sequence is said to be -summable to if and is denoted by
Definition 3. A double sequence is said to be strongly -summable to if is -summable to zero. In this case, we write . The set of all strongly -summable sequences is denoted by asBy considering the matrix in (3) for any , natural density and statistical convergence can be defined as follows.
Definition 4 (-density). Let be a subset of . Then, -density of is denoted by and defined byif the limit exists.
Definition 5 (-statistical convergence). A double sequence is said to be -statistically convergent to if for every , -density of the set is zero, i.e.,It is denoted by . The set of all -statistically convergent sequences is denoted by , i.e.,Let and be sequences of positive natural numbers and and . Take , where withIf we consider as in (10), then it is clear that -statistical convergence coincides weighted statistical convergence (or ) which was defined and studied by Çınar and Et in [29].
2. Main Results and Proofs
In this section, we will give some properties of -statistical convergence and comparison with strong -summability. Moreover, inclusion results for are given.
Theorem 1. Let and be two double sequences. Then,(i)If and , then (ii)If and , then
Proof. Omitted.
We define each of the following sets:
Theorem 2. Let . If and , then .
Proof. Assume that , , and . Take any , and denote the setsSince and , we can write and . It is clear that inclusionholds. Therefore, the inequalityholds. Also, the sets and are disjoint andFrom (15), we obtainThe last equality gives that , but this is a contradiction to . So, -statistical limit of is unique.
Theorem 3. Let be a sequence from . Then, the following statements hold true:(i)If , then and (ii)If and , then
Proof. (i) Let and . In this case, we obtain(ii)After taking the limit when in the above inequality, we obtain .(iii)Assume that and . Then, there exists a positive constant such thatfor all .
For a given , the following inequality holds:If we take the limit as , then we obtain that .
Remark 1. The converse of (i) in Theorem 3 does not hold.
Let us consider and as follows:Therefore, we havebut the following inequality holds:This gives that is strict.
Corollary 1. If , then for any .
Proof. Since and is regular for any , then . Therefore, Theorem 3 (i) gives that .
For , let us consider the seriesThe series and are convergent for all .
Theorem 4. For any , there exists a double sequence such that and .
Proof. For and , let us consider the sequence asfor . Let us show that . Since , we haveLet and be the and transformations of , respectively.
is also the -st transformation of , whereFrom (25), we have the following equality:Since , as , we obtain . This implies that .
By using the same argument, we can get .
Remark 2. Theorem 4 is valid for any .
Theorem 5. For any two , the methods and are consistent.
Proof. Let be a sequence such that and . If we consider the sequence as (25) in Theorem 4, we obtain and . From the uniqueness of the limit, we obtain that .
Theorem 6. Let . Then, if and only if and .
Proof. Let and be the - and -transforms of the sequence and , respectively.
If and , we obtainSimilarly, if and ,Then, it is clear from (24) thatfor all and .
It follows from the above equality and (23) and (24) thatSo, we obtainThen,whereThen, for the last discussion, we have if and only if the matrix is regular. So, we obtainThat is, . Also, we havei.e., .
The proof is completed.
Corollary 2. Let . Then, if and only if , , and .
Definition 6 (see [25]). Two sequences and in are said to be equivalent ifand it is denoted by .
Theorem 7. Let and be sequences in such that . Then, .
Proof. Let be an arbitrary sequence such thatfor any . Therefore,Since , for any fixed , there exists and such thatholds for all and . Hence, the following inequality holds:whenNow, by taking the limit as , we obtain that . We conclude that . Converse of this inclusion result can be obtained by the same way. Hence, the proof is completed.
Corollary 3. Let and be associated sets of . If is finite, then .
Theorem 8. Let be a sequence and . Then, implies if and only if for and .
Proof. Let for any . Since , we haveIf we take the limit as , the desired result is obtained if and only if .
Theorem 9. Let be a sequence and . Then, implies if and only if the sequence for and .
Proof. For any , let . Then, the inequalityholds. This gives the proof.
On the contrary, let us recall that is the space of all continuous real-valued functions on any compact subset of the real two-dimensional space. We know that is a Banach space with normSuppose that is a linear operator from into . It is clear that if implies , then the linear operator is positive on . Also, we denote the value of at a point by or only . The classical Korovkin approximation theorem (see [35]) was extended from single sequences to double sequences [36].
Now, we give the following theorem to prove the approximation theorem.
Theorem 10. Let be a double sequence of positive linear operators from into . Then,for all if and only ifwhere , , , , and .
Proof. Since each , assertion (48) follows immediately from assertion (47). Assume that (48) holds. Since , we have , where . Using the continuity of on for every , there is such that for all satisfying and . Then, we getAlso, by the linearity and positivity of the operators and from (49), we obtainwhere and . Then, taking , we obtainwhereSimilarly, we obtainWe now replace byWe choose such that for a given . Then, define the sets. It is clear that , and so, . Therefore, using condition (53), we obtainThis completes the proof.
Corollary 4. Let be a double sequence of positive linear operators from into . Then,for all if and only ifwhere , , , , and .
Remark 3. We now show in the following an example of a sequence of positive linear operators of two variables satisfying the conditions of Theorem 10 but does not satisfy the conditions of the Korovkin theorem. Consider the following Bernstein operators:where .
Let , , and . Then, by Corollary 4, we obtainfor all . with , whereLet . The double sequence is neither -convergent nor statistically convergent, but is statistically summable to zero. , , , and , and the double sequence satisfies condition (48) for . Hence, we getWe have since , and hence,It is easy to see that does not satisfy the conditions of the classical Korovkin-type theorem since and do not exist; this proves the claim.
3. Conclusion
In this paper, we introduce -statistical convergence for double sequences and give the inclusion results for different ’s. These new results can be viewed as a generalization of previously known results. The new concept can be applied to the approximation theory, Fourier analysis, topology, and so on. The -statistically Cauchy sequence can be described, and its properties can be studied. Also, ideal convergent sequence spaces can be given.
Data Availability
All the data in the manuscript are available upon request to the corresponding author.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors of the manuscript read and agreed to its content and were accountable for all aspects of the accuracy and integrity of the manuscript.