Abstract and Applied Analysis

Volume 2013 (2013), Article ID 264520, 5 pages

http://dx.doi.org/10.1155/2013/264520

Research Article

-Statistical Convergence in Paranormed Space

Mohammed A. Alghamdi¹ and Mohammad Mursaleen²

¹Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia

²Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

Received 24 October 2013; Accepted 9 December 2013

Academic Editor: S. A. Mohiuddine

Copyright © 2013 Mohammed A. Alghamdi and Mohammad Mursaleen. 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.

The concept of -statistical convergence for sequences of real numbers was introduced in Mursaleen (2000). In this paper, we prove decomposition theorem for -statistical convergence. We also define and study -statistical convergence, -statistically Cauchy, and strongly -summability in Paranormed Space.

1. Introduction

The notion of statistical convergence was first introduced by Fast [1]. In the recent years, statistical summability became one of the most active areas of research in summability theory, which was further generalized as lacunary statistical convergence [2], -statical convergence [3], statistical -summability [4], and statistical -convergence [5]. Maddox [6] studied this notion in locally convex Hausdorff topological spaces and Kolk [7] defined and studied this notion in Banach spaces while Çakalli [8] extended it to topological Hausdorff groups. The concept of statistical convergence is studied in probabilistic normed space and in intuitionistic fuzzy normed spaces in [9, 10]. Recently, the statistical convergence has been studied in Paranormed Space and locally solid Riesz spaces in [11, 12], respectively. Therefore, one can choose either some different setup to study these concepts or generalizing the existing concepts through different means. In this paper, we will study the concept of -statistical convergence, -statistical Cauchy, and strongly  -summability in Paranormed Space.

A paranorm is a function defined on a linear space such that for all

if ,,,if is a sequence of scalars with    and , with    in the sense that   , then   , in the sense that   .

A paranorm for which implies that is called a total paranorm on , and the pair is called a total Paranormed Space.

2. λ-Statistical Convergence

Let be a nondecreasing sequence of positive numbers tending to such that

The generalized de la Vallée-Poussin mean is defined by

where .

A sequence is said to be -summable to a number if

Let be a subset of the set of natural numbers . Then, the -density of is defined as

The number sequence is said to be -statistically convergent to the number (c.f. [3, 13, 14]) if ; that is, if for each ,

In this case we write and we denote the set of all -statistically convergent sequences by . In case , -density reduces to the natural density and -statistical convergence reduces to statistical convergence. This notion for double sequences has been studied in [15].

A sequence is said to be strongly  -summable   to the limit   [14] if

and we write it as . In this case is called the -limit of .

The following relation was established in [14].

Theorem 1. If  and a sequence is strongly  -summable to , then it is -statistically convergent to . If a bounded sequence is -statistically convergent to , then it is strongly  -summable to .

The following theorem is -statistical version of Connor’s Decomposition Theorem [16].

Theorem 2. If is strongly  -summable or statistically -convergent to , then there is a convergent sequence and a -statistically null sequence such that is convergent to and

Moreover, if is bounded, then and both are bounded.

Proof. By Theorem 1, it follows that is -statistically convergent to if is strongly  -summable to . Set and choose a strictly increasing sequence of positive integers such that

for . Define and as follows.

If set and . Let and . Now we set

Clearly, and and are bounded, if is bounded. Also, we observe that for , we have

Hence, , since was arbitrary.

Next we observe that

for any natural number and . Hence, ; that is, is -statistically null.

We now show that if and such that , then for all . Recall from the construction that if , then only if . It follows that if , then

Consequently,

if and . That is,

This completes the proof of the theorem.

3. Application to Fourier Series

Let be a Lebesgue integrable function on the torus ; that is, . The Fourier series of is defined by

where the Fourier coefficients are defined by

The symmetric partial sums of the series in (15) are defined by

The conjugate series to the Fourier series in (15) is defined by [17, Vol. I, pp. 49]

Clearly, it follows from (15) and (18) that

and the power series

is analytic on the open unit disk , due to the fact that

The conjugate function of a function is defined by

in the “principal value” sense and that exists at almost every .

The following is -statistical version of [18] (c.f. [19, Theorem 2.1 (ii)]).

Theorem 3. If , then for any its Fourier series is strongly  -summable to at almost every . Furthermore, its conjugate series (18) is strongly  -summable for any to the conjugate function defined in (22) at almost every .

From Theorems 1 and 3, we easily get the following useful result.

Theorem 4. If , then its Fourier series is -statistically convergent to at almost every . Furthermore, its conjugate series (18) is -statistically convergent to the conjugate function defined in (22) at almost every .

4. λ-Statistical Convergence in Paranormed Space

Recently, statistical convergence, statistical Cauchy, and strongly  Cesàro summability have been studied in Paranormed Space by Alotaibi and Alroqi [11].

In this paper, we define and study the notion of -summable, -statistical convergence, -statistical Cauchy, and strongly  -summability in Paranormed Space.

Let be a Paranormed Space.

A sequence is said to be convergent to the number in if, for every , there exists a positive integer such that whenever . In this case, we write -, and is called the -limit of .

We define the following.

Definition 5. A sequence is said to be -statistically convergent to the number in if, for each ,

In this case we write -.

Definition 6. A sequence is said to be -statistically Cauchy sequence in if for every there exists a number such that

Definition 7. A sequence is said to be strongly  -summable   to the limit   in if

and we write it as . In this case is called the -limit of .

Now we define another type of convergence in Paranormed Space.

Definition 8. A sequence in a Paranormed Space is said to -convergent to if there exists an index set , , with such that . In this case, we write .

First we prove the following results on -statistical convergence in .

Theorem 9. If -, then - but converse need not be true in general.

Proof. Let -. Then, for every , there is a positive integer such that

for all . Since the set is finite, . Hence, -.

The following example shows that the converse need not be true.

Example 10. Let with the paranorm . Define a sequence by

and write

We see that

and hence

Therefore - does not exist. On the other hand ; that is, -.

This completes the proof of the theorem.

We can easily prove the following results on -statistical convergence in similar to those of [11].

Theorem 11. If a sequence is -statistically convergent in , then -limit is unique.

Theorem 12. Let - and . Then,

(i)-,(ii)- .

Theorem 13. Let be a complete Paranormed Space. Then a sequence of points in is -statistically convergent if and only if it is -statistically Cauchy.

Theorem 14. (a) If and , then is -statistically convergent to in .

(b) If is bounded and -statistically convergent to in , then .

Theorem 15. Let be a complete Paranormed Space. Then a sequence of points in is -statistically convergent if and only if it is -statistically Cauchy.

Note that the proof of Theorem 2.4 [11] is incorrect and the correct proof is given in the following theorem which is generalization of Theorem 2.4 [11]. Another form of this result is given in [20] for ideal convergence.

Theorem 16. A sequence in is -statistically convergent to if and only if it is -convergent to .

Proof. Suppose that is -statistically convergent to ; that is, -. Now, write for .

Then ,

Now we have to show that, for , is -convergent to . On contrary suppose that is not -convergent to . Therefore, there is such that for infinitely many terms. Let and , .

Then

and by (32), . Hence , which contradicts (33) and we get that is -convergent to . Hence, is -convergent to .

Conversely, suppose that is -convergent to . Then there exists a set with such that -. Therefore, there is a positive integer such that for . Put and . Then and which implies that . Hence is -statistically convergent to ; that is -.

This completes the proof of the theorem.

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. (130-073-D1434). The authors, therefore, acknowledge with thanks DSR technical and financial support.

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