On lacunary statistical boundedness
© Bhardwaj et al.; licensee Springer. 2014
Received: 27 February 2014
Accepted: 30 July 2014
Published: 21 August 2014
A new concept of lacunary statistical boundedness is introduced. It is shown that, for a given lacunary sequence , a sequence is lacunary statistical bounded if and only if for ‘almost all k w.r.t. θ’, the values coincide with those of a bounded sequence. Apart from studying various algebraic properties and computing the Köthe-Toeplitz duals of the space of all lacunary statistical bounded sequences, a decomposition theorem is also established. We characterize those θ for which . Finally, we give a general description of inclusion between two arbitrary lacunary methods of statistical boundedness.
MSC:40C05, 40A05, 46A45.
Keywordslacunary sequence statistical boundedness statistical convergence Köthe-Toeplitz duals
1 Introduction and 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 , who called it ‘almost convergence’. Formally the concept of statistical convergence was introduced by Fast  in the year 1951 and later reintroduced by Schoenberg  in the year 1959. Statistical convergence also arises as an example of ‘convergence in density’ as introduced by Buck .
Although statistical convergence was introduced over nearly last 60 years, it has become an active area of research in recent years. Statistical convergence has been studied most recently by several authors [5–25].
The standard definition of ‘ is convergent to L’ requires that the set should be finite for every , where ℕ is the set of natural numbers.
where denotes the number of elements of K not exceeding n. A set K is said to be statistically dense  if . A subsequence of a sequence is said to be statistically dense if the set of all indices of its elements is statistically dense. Obviously, we have provided that K is a finite set of positive integers.
We shall be particularly concerned with those subsets of ℕ which have natural density zero. To facilitate this, Fridy  introduced the following notation: if is a sequence such that satisfies property P for all k except a set of natural density zero, then we say that satisfies P for ‘almost all k’ and we abbreviate this by ‘a.a.k’ .
Using this notation, we have the following.
Following Freedman et al. , by a lacunary sequence , where , we shall mean an increasing sequence of non-negative integers with as . The intervals determined by θ will be denoted by , and we let . Sums of the form will be written for convenience as and the ratio will be denoted by .
Fridy and Orhan  introduced and studied a concept of convergence, called lacunary statistical convergence, that is related to statistical convergence in the same way as is related to .
Definition 1.2 Let θ be a lacunary sequence. The number sequence is lacunary statistical convergent or -convergent to L provided that for every , . In this case, we write or , and we define .
where is the characteristic function of K.
For , the Cesàro mean of order one, reduces to , i.e., the natural density of the set K.
then reduces to , i.e., the lacunary density or θ density of K. Thus the lacunary density or θ density is a particular case of the A-density.
We now introduce the following notation.
For a given lacunary sequence , if is a sequence such that satisfies property P for all k, except a set of θ density zero, then we say that satisfies P for ‘almost all k with respect to θ’ and we abbreviate this by ‘a.a.k w.r.t. θ’.
Using this notation, we have the following.
In 1997, Fridy and Orhan  introduced the concept of statistical boundedness as follows.
We denote the set of all statistically bounded sequences by .
In the same year, i.e., 1997, Tripathy  proved a decomposition theorem for statistically bounded sequences and also established a necessary and sufficient condition for a sequence to be statistically bounded.
Quite recently, Bhardwaj and Gupta  have introduced and studied the concepts of statistical boundedness of order α, λ-statistical boundedness and λ-statistical boundedness of order α.
The main object of this paper is to introduce and study the new concept of lacunary statistical boundedness.
For a given lacunary sequence , by we denote the set of all -bounded sequences. Obviously, is a linear space with respect to co-ordinatewise addition and scalar multiplication.
In the next section we establish elementary relations among the concepts of boundedness, lacunary statistical boundedness and lacunary statistical convergence of sequences of numbers. It is shown that for a given lacunary sequence , a sequence is lacunary statistical bounded if and only if for almost all k w.r.t. θ, the values coincide with those of a bounded sequence. Apart from studying various algebraic properties and computing the Köthe-Toeplitz duals of the sequence space , a decompositon theorem for lacunary statistical boundedness is also established. In Section 3, we characterize those θ for which . It is also shown that precisely those subsequences of a lacunary statistical bounded sequence are lacunary statistical bounded which are lacunary statistical dense. In the last section, we consider the inclusion of by where is a lacunary refinement of θ. Recall  that the lacunary sequence is called a lacunary refinement of the lacunary sequence if . A general description of inclusion between two arbitrary lacunary methods of statistical boundedness is also given.
normal (or solid) if whenever , , for some ,
monotone if it contains the canonical preimages of all its stepspaces,
sequence algebra if whenever .
Obviously , where ϕ is the well-known sequence space of finitely non-zero scalar sequences. Also if , then for or β. For any sequence space X, we denote by where or β. It is clear that where or β.
For a sequence space X, if then X is called a Köthe space or a perfect sequence space.
2 Some inclusion relations and a decomposition theorem
We begin by establishing elementary connections between boundedness, lacunary statistical boundedness and lacunary statistical convergence.
We state the following result without proof in view of the fact that the empty set has zero lacunary density for every lacunary sequence θ.
Theorem 2.1 Every bounded sequence is lacunary statistical bounded, i.e., for every lacunary sequence θ.
Remark 2.2 The converse of the above theorem need not be true.
where is a lacunary sequence. Clearly . But and so .
Remark 2.4 From Theorem 2.1 and above example, it is clear that lacunary statistical boundedness is a generalization of the usual concept of boundedness of sequences.
Theorem 2.5 Every lacunary statistical convergent sequence is lacunary statistical bounded, but not conversely.
Proof Let . Then for each , we have . The result now follows from the fact that .
It can easily be verified that lacunary statistical bounded sequence is not lacunary statistical convergent. □
Remark 2.6 There are sequences which are not lacunary statistical bounded for any lacunary sequence θ. One example of such sequences is where for each . Hence where ω is the space of all scalar sequences.
Theorem 2.7 For a given lacunary sequence , a sequence is lacunary statistical bounded if and only if there exists a bounded sequence such that a.a.k w.r.t. θ.
Then and a.a.k w.r.t. θ.
Conversely, as so there exists such that for all . Let . As , so a.a.k w.r.t. θ. □
Corollary 2.8 Every lacunary statistical bounded sequence has a bounded subsequence.
A decomposition theorem for statistical boundedness was given by Tripathy . We now give a lacunary analog of this result.
Theorem 2.9 (Decomposition theorem)
If is a lacunary statistical bounded sequence, then there exists a bounded sequence and a lacunary statistical null sequence such that . However, this decomposition is not unique.
Clearly where y is a bounded sequence and z is a lacunary statistical null sequence, i.e., where denotes the set of all lacunary statistical null sequences. As , so . Consequently, we have . Using the fact that , where ϕ is the space of finitely non-zero scalar sequences, we have , i.e., the decomposition is not unique. □
is normal and hence monotone.
is a sequence algebra.
The proof is easy and so omitted.
Proposition 2.11 , the space of finitely non-zero scalar sequences.
Proof To show that , it is sufficient to show that since obviously. Let . Then for all . Suppose , i.e., has infinitely many non-zero terms. Following Lemma 5 of , for each , if contains a k such that , let be the least such k; otherwise leave undefined. Thus there are infinitely many ’s and . Now define if for some and otherwise. Now as and so . But for infinitely many r and so . □
In view of the fact [, p.52] that for a monotone sequence space, α- and β-dual spaces coincide, we have .
Corollary 2.12 is not perfect.
Proof As , so is not a perfect space. □
3 Lacunary statistical boundedness versus statistical boundedness
In this section, we study the inclusions and under certain restrictions on and characterize those θ for which .
Lemma 3.1 For any lacunary sequence θ, if and only if .
Proof (Sufficiency). If , then there exists such that for sufficiently large r. Since , so we have and . For , there exists such that .
This proves the sufficiency.
i.e., for all . Also, for , . Thus .
Since implies , we have . □
Remark 3.2 The sequence , constructed in the necessity part of above lemma, is an example of a statistical bounded sequence which is not lacunary statistical bounded.
Lemma 3.3 For any lacunary sequence θ, if and only if .
and this is true for all . Thus . □
Remark 3.4 The sequence , constructed in the necessity part of above lemma, is an example of a lacunary statistical bounded sequence which is not statistical bounded.
Combining Lemma 3.1 and Lemma 3.3 we have the following.
Theorem 3.5 Let θ be a lacunary sequence. Then if and only if .
Proof In view of Lemma 3.1, we have . Suppose, if possible, but . We have for all with . If we take , then in view of Theorem 3.5 we have and so , contrary to our supposition. Hence . The remaining part can be proved similarly and hence is omitted. □
Remark 3.7 It is well known that every subsequence of a bounded sequence is bounded. However, for lacunary statistical bounded sequences this is no longer true. This can be verified by the following example.
i.e., . Then . However, is a subsequence of the lacunary statistical bounded sequence but is not lacunary statistical bounded.
We now characterize those subsequences of a lacunary statistical bounded sequence which are themselves lacunary statistical bounded. Before doing so, we introduce the following definition.
Definition 3.9 Let θ be a lacunary sequence. A subset A of ℕ is said to be lacunary statistical dense or θ dense if . A subsequence of a sequence is said to be lacunary statistical dense if the set of all indices of its elements is lacunary statistical dense.
In view of Theorem 2.1 of Burgin and Duman  we state the following result without proof.
Theorem 3.10 A sequence is lacunary statistical bounded if and only if every lacunary statistical dense subsequence of it is lacunary statistical bounded.
4 Inclusion between two lacunary methods of statistical boundedness
Our first result shows that if β is a refinement of θ, then .
We state the following result, which can be established following the technique of Theorem 7 of Fridy and Orhan .
Theorem 4.1 If β is a lacunary refinement of θ and , then , i.e., .
Following Li , we state the next result without proof in which we impose certain restriction on a refinement β of θ so as to have the reverse inclusion i.e., .
Theorem 4.2 Suppose is a lacunary refinement of the lacunary sequence . Let and , . If there exists such that for every , then .
The following result is a consequence of Theorem 4.1 and Theorem 4.2.
Corollary 4.3 Under the hypothesis of Theorem 4.2, we have .
The next theorem provides a sufficient condition for lacunary sequences and to yield the inclusion relation .
In view of Theorem 2 of Li  we state the following result without proof.
Theorem 4.4 Suppose , are two lacunary sequences. Let , , , and , . If there exists such that for every , provided , then .
Remark 4.5 If the condition in Theorem 4.4 is replaced by for every , provided , then it can be seen that .
Combining Remark 4.5 and Theorem 4.4, we get the following.
Theorem 4.6 Suppose , are two lacunary sequences. Let , , , and , . If there exists such that for every , provided , then .
The authors are grateful to the referee for his/her valuable comments and suggestions, which have improved the presentation of the paper.
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