Density by Moduli and Lacunary Statistical Convergence

Bhardwaj, Vinod K.; Dhawan, Shweta

doi:https://doi.org/10.1155/2016/9365037

Abstract and Applied Analysis

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Volume 2016 | Article ID 9365037 | https://doi.org/10.1155/2016/9365037

Density by Moduli and Lacunary Statistical Convergence

Vinod K. Bhardwaj¹and Shweta Dhawan²

Academic Editor: Feyzi Başar

Received15 Nov 2015

Accepted05 Jan 2016

Published15 Feb 2016

Abstract

We have introduced and studied a new concept of -lacunary statistical convergence, where is an unbounded modulus. It is shown that, under certain conditions on a modulus , the concepts of lacunary strong convergence with respect to a modulus and -lacunary statistical convergence are equivalent on bounded sequences. We further characterize those for which , where and denote the sets of all -lacunary statistically convergent sequences and -statistically convergent sequences, respectively. A general description of inclusion between two arbitrary lacunary methods of -statistical convergence is given. Finally, we give an -analog of the Cauchy criterion for convergence and a Tauberian theorem for -convergence is also proved.

1. Introduction and Historical Background

Statistical convergence is a generalization of the usual notion of convergence. The idea of statistical convergence was given in the first edition (published in Warsaw in 1935) of the monograph of Zygmund [1], who called it “almost convergence.” Formally the concept of statistical convergence was introduced by Fast [2] and Steinhaus [3] independently in the year 1951 and later reintroduced by Schoenberg [4] in the year 1959.

Although statistical convergence was introduced over nearly the last sixty years, it has become an active area of research in recent years. Statistical convergence has been studied by several authors [5–14].

Let denote the set of all natural numbers. A number sequence is said to be statistically convergent to the number if for each the set has natural density zero, where the natural density of a subset (chapter 11, see [15]) is defined bywhere denotes the number of elements of not exceeding . Obviously we have provided that is a finite set of positive integers. If a sequence is statistically convergent to , then we write it as or . The set of all statistically convergent sequences is denoted by .

The notion of modulus function was introduced by Nakano [16] in 1953. Following Ruckle [17] and Maddox [18], we recall that a modulus is a function from to such that(i) if and only if ,(ii) for ,(iii) is increasing,(iv) is continuous from the right at . Because of (ii), so that, in view of (iv), is continuous everywhere on . A modulus may be unbounded or bounded. For example, , where , is unbounded, but is bounded.

Connor [11], Öztürk and Bilgin [19], Ghosh and Srivastava [20], Bhardwaj and Singh [21–23], Çolak [24], Altin and Et [25], Aizpuru et al. [5], Bhardwaj and Dhawan [6], Bhardwaj et al. [26], and some others have used a modulus function to construct some sequence spaces.

In the year 2014, Aizpuru et al. [5] defined a new concept of density with the help of an unbounded modulus function and, as a consequence, they obtained a new concept of nonmatrix convergence, namely, -statistical convergence, which is intermediate between the ordinary convergence and the statistical convergence and agrees with the statistical convergence when the modulus function is the identity mapping.

Quite recently, Bhardwaj and Dhawan [6] and Bhardwaj et al. [26] have introduced and studied the concepts of -statistical convergence of order and -statistical boundedness, respectively, by using the approach of Aizpuru et al. [5].

We now recall some definitions that will be needed in the sequel.

Definition 1 (see [5]). Let be an unbounded modulus function. The -density of a set is defined by in case this limit exists. Clearly, finite sets have zero -density and does not hold, in general. But if then . Note that Aizpuru et al. [5] have used the notation to denote the -density of a set .

Remark 2. For any unbounded modulus and , implies that . But converse need not be true in the sense that a set having zero natural density may have nonzero -density with respect to some unbounded modulus . For example, if we take and , then but .

Definition 3 (see [5]). Let be an unbounded modulus function. A number sequence is said to be -statistically convergent to or -convergent to , if, for each , and one writes it as or . The set of all -statistically convergent sequences is denoted by .

In view of Definition 3 and Remark 2, it follows that every -statistically convergent sequence is statistically convergent, but a statistically convergent sequence need not be -statistically convergent for every unbounded modulus .

By a lacunary sequence , where , we will mean an increasing sequence of nonnegative integers with as . The intervals determined by will be denoted by , and we let . The ratio , which also occurs frequently, will be denoted by .

The space of all lacunary strongly convergent sequences, , was defined by Freedman et al. [27] as follows:If is lacunary strongly convergent to , then we write or .

There is a strong connection [27] between and the space of strongly Cesàro summable sequences, which is defined by In the special case, where , we have . In fact, for a lacunary sequence , if and only if ([27], on page 511).

Note that the space of strongly Cesàro summable sequences was denoted by in [27].

In the year 1986, Maddox [28] extended the concept of strong Cesàro summability to that of strong Cesàro summability with respect to a modulus . A sequence is said to be strongly Cesàro summable with respect to a modulus to if The space of strongly Cesàro summable sequences, with respect to a modulus , is denoted by .

Further, in the year 1994, the notion of lacunary strong convergence was extended to that of lacunary strong convergence with respect to a modulus by Pehlivan and Fisher [29]. The space of lacunary strongly convergent sequences with respect to a modulus is defined as If is lacunary strongly convergent with respect to a modulus to , then we write or .

Remark 4. We will denote the spaces and by and throughout the paper.

Fridy and Orhan [12] introduced the concept of lacunary statistical convergence as follows.

Definition 5. Let be a lacunary sequence. A number sequence is said to be lacunary statistically convergent to or -convergent to , if, for each , In this case, one writes or . The set of all lacunary statistically convergent sequences is denoted by .

Throughout this paper , , and will denote the spaces of all, bounded, and convergent sequences of complex numbers, respectively. Moreover, we will be concerned only with the sequences of scalars.

Following a very recent and new approach of Aizpuru et al. [5], we first introduced the notion of -density and, consequently, obtained a new concept of nonmatrix convergence, namely, -statistical convergence of order [6], where is an unbounded modulus function and . The -density of the subset of is defined as follows.

Definition 6. Let be an unbounded modulus function and let be any real number such that . Then in case this limit exists, where denotes the number of elements of not exceeding .

Now, in this paper we introduce a new concept of -lacunary statistical convergence, where is an unbounded modulus, as follows.

Definition 7. Let be an unbounded modulus and let be a lacunary sequence. A sequence is said to be -lacunary statistically convergent to or -convergent to , if, for each , In this case, one writes or .

For a given lacunary sequence and an unbounded modulus , by we denote the set of all -lacunary statistically convergent sequences.

Remark 8. In case , the concept of -lacunary statistical convergence reduces to that of lacunary statistical convergence.

In the next section it is shown that the concept of -lacunary statistical convergence is intermediate between the ordinary convergence and the lacunary statistical convergence. It is also proved that, under certain conditions on a modulus , if a sequence is lacunary strongly convergent with respect to a modulus then it is -lacunary statistically convergent and that the concepts of lacunary strong convergence with respect to a modulus and -lacunary statistical convergence are equivalent on bounded sequences. In Section 3, we characterize those for which , where is an unbounded modulus such that and there is a positive constant such that , for all . Note that it was shown by Maddox [18] that, for any modulus , exists. Maddox [28] also showed that there exists an unbounded modulus for which there is a positive constant such that , for all , . In Section 4, we first observe that it is possible for a sequence to have different -limits for different and . In Theorem 24, we investigate certain conditions under which this situation cannot occur. In Section 5, a general description of inclusion between two arbitrary lacunary methods of -statistical convergence is given. In the last section, -lacunary statistical analog of Cauchy criterion for convergence is established. A Tauberian theorem for -convergence is also given.

2. -Lacunary Statistical Convergence

We begin by establishing elementary connections between convergence, -lacunary statistical convergence, and lacunary statistical convergence.

Theorem 9. Every convergent sequence is -lacunary statistically convergent; that is, for any unbounded modulus and lacunary sequence .

Proof. Let be any convergent sequence. Then, for each , the set is finite. Suppose . Now, since and being modulus is increasing, therefore Taking limit as , on both sides, we get

Remark 10. The inclusion is strict as the sequence given in the necessity part of Lemma 19 is an example of a sequence which is -convergent but not convergent.

Theorem 11. Every -lacunary statistically convergent sequence is lacunary statistically convergent.

Proof. Suppose is -lacunary statistically convergent to . Then by the definition of limit and the fact that being modulus is subadditive, for every , there exists such that, for , we have and since is increasing, we have Hence, is lacunary statistically convergent to .

Remark 12. It seems that the inclusion is strict. But right now we are not in a position to give an example of a sequence which is -convergent but not -convergent. So it is left as an open problem.

Remark 13. From Theorems 9 and 11, we can say that the concept of -lacunary statistical convergence is intermediate between the usual notion of convergence and the lacunary statistical convergence.

We now establish a relationship between -lacunary statistical convergence and lacunary strong convergence with respect to a modulus . Fridy and Orhan [12] showed that on bounded sequences the concept of lacunary statistical convergence is equivalent to lacunary strong convergence. We now wish to find some condition on , if any, so that the concept of -lacunary statistical convergence becomes equivalent to lacunary strong convergence with respect to a modulus .

Theorem 14. Let be a lacunary sequence; then consider the following: (a)For any unbounded modulus for which and there is a positive constant such that , for all , ,(i) implies ,(ii) is a proper subset of .(b) and imply , for any unbounded modulus .(c) for any unbounded modulus for which and there is a positive constant such that for all , .

Proof. (a) (i) For any sequence and , by the definition of a modulus function (ii) and (iii) we have from where it follows that as and .
(ii) To show the strictness of inclusion, consider the sequence such that is to be at the first integers in , and otherwise. Note that is not bounded. Also, for every , because, are positive and
Thus, . On the other hand, as, , and are positive. Hence .
(b) Suppose that and ; say for all . Given , we have Taking limit on both sides as , we get in view of Theorem 11 and the fact that is increasing.
(c) This is an immediate consequence of (a) and (b).

Remark 15. The example given in part (a) of the above theorem shows that the boundedness condition cannot be omitted from the hypothesis of part (b).

Remark 16. If we take in Theorem 14, we obtain Theorem 1 of Fridy and Orhan [12].

3. -Lacunary Statistical Convergence versus -Statistical Convergence

In this section we study the inclusions and under certain restrictions on and .

Lemma 17. For any lacunary sequence and unbounded modulus for which and there is a positive constant such that , for all , , one has if and only if .

Proof.
Sufficiency. If , then there exists such that for sufficiently large . Since , we have for sufficiently large . If , then, for given and sufficiently large , we have This proves the sufficiency.
Necessity. Assume that . Proceeding as in Lemma of [27], we can select a subsequence of satisfying Define a bounded sequence byIt is shown in Lemma of [27] that but . Thus in view of Theorem of [29] and Theorem 14 we have . On the other hand, it follows from Theorem of [28] and Corollary of [6] that . Hence . But this is a contradiction to the assumption that . This contradiction shows that our assumption is wrong. Hence .

Remark 18. The sequence , constructed in the necessity part of the above lemma, is an example of -statistically convergent sequence which is not -lacunary statistically convergent.

Lemma 19. For any lacunary sequence and unbounded modulus for which and there is a positive constant such that , for all , one has if and only if .

Proof.
Sufficiency. If , then there is such that for all . Now, suppose that and . Therefore, for given , there exists such that for all Let . Using this notation, we haveNow, let and let be an integer such that , and then we can write from where the sufficiency follows immediately, in view of the fact that .
Necessity. Suppose that . Following Lemma of [27], we can select a subsequence of lacunary sequence such that . Define a bounded sequence byIt is shown in [27], on page , that but . By Theorem of [29] and Theorem 14, we conclude that , but , in view of Theorem of [10] and the fact that every -statistically convergent sequence is statistically convergent. Hence . But this is a contradiction to the assumption that . This contradiction shows that .

Remark 20. The sequence , constructed in the necessity part of the above lemma, is an example of -lacunary statistically convergent sequence which is not -statistically convergent.

Combining Lemmas 17 and 19 we have the following.

Theorem 21. For any lacunary sequence and unbounded modulus for which and there is a positive constant such that , for all , , one has if and only if .

Theorem 22. For any lacunary sequence and unbounded modulus for which and there is a positive constant such that , for all , , one has

Proof. In view of Lemma 17, we have . Suppose if possible but . We have for all for which . If we take , then, in view of Theorem 21, we have and so , contrary to our assumption. Hence . The remaining part can be proved similarly and hence is omitted.

Remark 23. The sequence constructed in part (a) of Theorem 14 belongs to for every lacunary sequence , as well as unbounded modulus for which and there is a positive constant such that for all . Hence .

4. Uniqueness of -Limit

For any fixed , the -limit is unique; however, Fridy and Orhan [12] showed that it is possible for a sequence to have different -limits for different ’s. They showed that this situation cannot occur if the sequence is statistically convergent. We now establish a similar result in case of -convergence. First we observe that it is possible for a sequence to have different -limits for different and . To illustrate this, let us take , , , and . Consider the sequence given in [27], in the proof of Theorem , for which and . Now, by applying Theorem of [29] and Theorem 14 part (a), we see that and .

In the next theorem we investigate certain conditions under which this situation cannot occur.

Theorem 24. For any two lacunary sequences and , if and , then , where and are unbounded modulus functions for which

To prove this theorem we need the following lemma.

Lemma 25. For any lacunary sequence , if , then , where is an unbounded modulus function for which

Proof. Suppose and , and . Let be such that . Using the definition of a modulus (iii) and (ii), we have Taking limit as on both sides, we getNow consider the th term of the sequence :Also, in view of the choice of unbounded modulus , we haveNow, using (33) in (32), we havewhere because . Since is a lacunary sequence and being modulus is increasing, the term on the right hand side of (34) is a regular weighted mean transformation of , and therefore it, too, tends to zero as . Thus Also, since is a subsequence of sequence , we conclude thatBut this is a contradiction to (31). This contradiction shows that .

Proof of Theorem 24. By Lemma 25, we haveBut, according to Corollary of Aizpuru et al. [5], we haveTherefore, from (37) and (38) we have

If we take in Theorem 24 we have the following.

Corollary 26. For any two lacunary sequences and , if and , then , where is an unbounded modulus function such that

If we take in Corollary 26 we obtain the following result which contains Theorem of Fridy and Orhan [12].

Corollary 27. For any two lacunary sequences and , if and , then .

If we take in Theorem 24 we have the following.

Corollary 28. For any lacunary sequence , if and , then , where and are unbounded modulus functions for which

5. Inclusion between Two Lacunary Methods of -Statistical Convergence

Our first result shows that, for certain , if is a lacunary refinement of , then . To establish this result, we first recall the definition of a lacunary refinement of lacunary sequence [27].

Definition 29. The lacunary sequence is called a lacunary refinement of the lacunary sequence if .

Theorem 30. If is a lacunary refinement of and is an unbounded modulus such that then implies .

Proof. Suppose each of contains the points of so that Note that, for all , because . Let . Therefore, for each , we have where and , whenceFor each , we haveAlso, in view of the choice of unbounded modulus and using the fact that is increasing, we haveUsing (47) in (46), we havewhere . Since the term on the right hand side of (48) is a regular weighted mean transformation of , which tends to zero as , therefore the term on the right hand side of (48) also tends to zero as . Thus,Hence .

Our next result deals with the reverse inclusion.

Theorem 31. Let be an unbounded modulus and is a lacunary refinement of the lacunary sequence . Let , , , and , , . If there exists such that then implies .

Proof. For any , and every , we can find such that ; then we have from where it follows that .

In the next theorem we deal with a more general situation.

Theorem 32. Let and be any two unbounded modulus functions such that , for all , and is a lacunary refinement of the lacunary sequence . Let , , and , . If there exists such that then implies .

Proof. For any , and every , we can find such that ; then we have from where it follows that .

In the next theorem we show that the inclusion is possible even if none of and is refinement of the other.

Theorem 33. Let be an unbounded modulus such that Suppose and are two lacunary sequences. Let , , , , and . If there exists such thatprovided , where denotes the length of the interval , then implies .

The proof is similar to Theorem of Li [14] and hence is omitted.

Remark 34. If the condition in Theorem 33 is replaced by for every , provided , where denotes the length of the interval , it can be seen that implies .

Combining Remark 34 and Theorem 33, we get the following.

Theorem 35. Let be an unbounded modulus such thatSuppose and are two lacunary sequences. Let , , , , and . If there exists such thatprovided , where denotes the length of the interval , then .

6. The -Cauchy Criterion and a Tauberian Theorem

We now introduce the -analog of the Cauchy criterion, which turns out to be equivalent to the -convergence.

Definition 36. Let be a lacunary sequence and let be an unbounded modulus. The sequence is said to be -lacunary statistically Cauchy or -Cauchy sequence if there is a subsequence of such that , for each , , and, for every ,

Theorem 37. The sequence is -convergent if and only if is -Cauchy sequence, where is an unbounded modulus and is a lacunary sequence.

The proof is similar to proof of Theorem of Fridy and Orhan [13] and hence is omitted.

Corollary 38. Every -convergent sequence has a convergent subsequence.

This result leads to the following Tauberian theorem. We state the following result, which can be established following the technique of Theorem of Fridy and Orhan [13].

Theorem 39. If and then .

Conflict of Interests

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

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Copyright © 2016 Vinod K. Bhardwaj and Shweta Dhawan. 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.

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