pure and applied mathematics letters

P A M L December 2013

PURE AND APPLIED MATHEMATICS LETTERS An International Journal A characterization of weak statistical convergence via lacunary sequences and ideals

HCTM Technical Campus P A M L December 2 0 1 3

Pure and Applied Mathematics Letters 1(2013)25-33

Contents lists available at HCTM, Technical Campus

Pure and Applied Mathematics Letters Journal homepage: http:pamletters.org

A characterization of weak statistical convergence via lacunary sequences and ideals Meenakshi1, M. S. Saroa1 1

Department of Mathematics, Maharishi Markandeshwar University, Mullana Ambala, Haryana, India.

Abstract For any lacunary sequence 𝜃 = (𝑘𝑟 ) and an ideal 𝐼 ⊂ 𝑃(ℕ), we intend in present paper to define and study certain new generalizations of weak convergence which we call 𝑊𝑆(𝐼) −convergence and 𝑊𝑆𝜃 (𝐼) −convergence on a Banach space 𝑋. Later, notions of 𝑊𝑆𝜃 (𝐼) −limit points and 𝑊𝑆𝜃 (𝐼) −cluster points are also defined and some specific relations are obtained.

Keywords: weak convergence, statistical convergence, lacunary statistical convergence, ideal convergence.

AMS subject classification: 40A05; 40A35; 46B10; 46B99. Article Information: Received 07.07.2013

Revised 28.12.2013

Accepted 30.12.2013

Communicated by Prof. S. S. Bhatia

1. Introduction and Background Fast [8] and Schonenberg [23] independently introduced statistical convergence in order to assign limit to those sequences which are not convergent in the usual sense with the help of asymptotic density of subsets of ℕ, the set of positive integers. Definition 1.1 [8] A sequence 𝑥 = 𝑥𝑘 of numbers is said to be statistically convergent to a number 𝐿 provided that, for every 𝜀 > 0, 1 lim 𝑘 ≤ 𝑛: 𝑥𝑘 − 𝐿 ≥ 𝜀 = 0, 𝑛 𝑛 where {⋅} denotes the cardinality of the enclosed set. In this case, we write 𝑆 − lim𝑘→∞ 𝑥𝑘 = 𝐿. In past years, many problems arising naturally in analysis have been solved by use of statistical convergence. But the quick process of change came after the works of Šalát [22], Fridy [9] and Connor [3]. This concept has been discussed in many fields. For instance in trigonometric series [24], summability theory [12], intuitionistic fuzzy normed spaces [16], probabilistic normed spaces [15], locally convex spaces [20] and Banach spaces [17]. In [11], Fridy and Orhan generalized statistical convergence to lacunary statistical convergence with the help of a lacunary sequence. By a lacunary sequence, we mean an increasing sequence 𝜃 = (𝑘𝑟 ) of positive integers such that 𝑘0 = 0 and ℎ𝑟 = 𝑘𝑟 − 𝑘𝑟−1 → ∞ as 𝑟 → ∞. The intervals determined by 𝜃 will be denoted by 𝐼𝑟 = (𝑘𝑟−1 , 𝑘𝑟 ] and the ratio 𝑘𝑟 /𝑘𝑟−1 is denoted by 𝑞𝑟 . Definition 1.2 [11] For a lacunary sequence 𝜃 = 𝑘𝑟 , the sequence 𝑥 = 𝑥𝑘 of numbers is said to be lacunary statistically convergent to a number 𝐿 provided that, for every 𝜀 > 0, 𝛿𝜃 𝑘 ∈ 𝐼𝑟 : 𝑥𝑘 − 𝐿 ≥ 𝜀 where 𝛿𝜃 (𝐾) = lim𝑟

1 ℎ𝑟

= 0,

{𝑘 ∈ 𝐼𝑟 : 𝑘 ∈ 𝐾} . In this case, we write 𝑆𝜃 − lim𝑘 →∞ 𝑥𝑘 = 𝐿.

Definition 1.3 [11] A sequence 𝑥 = (𝑥𝑘 ) of numbers is said to be 𝑁𝜃 −convergent to 𝐿 provided that, for each 𝜀 > 0, lim 𝑟

1 ℎ𝑟

‍|𝑥𝑘 − 𝐿| = 0. 𝑘∈𝐼𝑟

In this case, we write 𝑁𝜃 − lim𝑘→∞ 𝑥𝑘 = 𝐿.

______________________ Corresponding author: Meenakshi, Mob. No. +91 8950651607, E-mail addresses: [email protected] 25

Connor 𝑒𝑡 𝑎𝑙. [5], have generalized statistical convergence to weak statistical convergence and given the description of Banach spaces with separable duals using weak statistical convergence. Let 𝑋 be a Banach space and (𝑥𝑘 ) be a sequence in 𝑋. Definition 1.4 [5] A sequence (𝑥𝑘 ) is said to be weak statistically convergent to 𝑥 ∈ 𝑋 provided that, for each 𝑓 ∈ 𝑋 ∗ ,(𝑋 ∗ is the continuous dual of X), each 𝜀 > 0, 𝛿 {𝑘 ≤ 𝑛: |𝑓(𝑥𝑘 − 𝑥)| ≥ 𝜀} = 0. In this case, we write 𝑊𝑆 − lim𝑘→∞ 𝑥𝑘 = 𝑥. Although, Bhardwaj and Bala [2] have introduced weak statistical Cauchy sequences in a normed space and weak statistical convergence in 𝑙𝑝 spaces. Bala [1], has generalized weak statistical convergence to weak* statistical convergence. In 2010, F. Nuray [21], has generalized weak statistical convergence to lacunary weak statistical convergence as follows. Definition 1.5 [21] A sequence (𝑥𝑘 ) is said to be weak lacunary statistically convergent to 𝑥 ∈ 𝑋 provided that, for each 𝑓 ∈ 𝑋 ∗ , each 𝜀 > 0, 𝛿𝜃 {𝑘 ∈ 𝐼𝑟 : |𝑓(𝑥𝑘 − 𝑥)| ≥ 𝜀} = 0. In this case, we write 𝑊𝑆𝜃 − lim𝑘→∞ 𝑥𝑘 = 𝑥. Definition 1.6 [21] A sequence

𝑥𝑘

is said to be weak 𝑁𝜃 −convergent to 𝑥 ∈ 𝑋 provided that, for each 𝑓 ∈ 𝑋 ∗ , each 𝜀 > 0, 1 ℎ𝑟

lim 𝑟

‍ 𝑓 𝑥𝑘 − 𝑥

=0

𝑘∈𝐼𝑟

We next give a brief description of 𝐼 −convergence. Kostyrko, Šalát and Wilczynski [18] have defined 𝐼 −convergence as a natural generalization of statistical convergence. For the set of positive integers ℕ, let 𝑃(ℕ) denotes the power set of ℕ. Definition 1.7 A family of sets 𝐼 ⊂ 𝑃(ℕ) is called an ideal in ℕ if and only if (i) (ii) (iii)

∅ ∈ 𝐼; For each 𝐴, 𝐵 ∈ 𝐼 we have 𝐴 ∪ 𝐵 ∈ 𝐼; For 𝐴 ∈ 𝐼 and 𝐵 ⊆ 𝐴 we have 𝐵 ∈ 𝐼.

Definition 1.8 A non-empty family of sets 𝐹 ⊂ 𝑃(ℕ) is called a filter on ℕ if and only if (i) (ii) (iii)

∅ ∉ 𝐹; For each 𝐴, 𝐵 ∈ 𝐹 we have 𝐴 ∩ 𝐵 ∈ 𝐹; For 𝐴 ∈ 𝐹 and 𝐵 ⊇ 𝐴 we have 𝐵 ∈ 𝐹.

An ideal 𝐼 is called non-trivial if 𝐼 ≠ 𝑃(ℕ). It immediately implies that 𝐼 ⊂ 𝑃(ℕ) is a non-trivial ideal if and only if the class 𝐹 = 𝐹(𝐼) = {ℕ − 𝐴: 𝐴 ∈ 𝐼} is a filter on ℕ. The filter 𝐹 = 𝐹(𝐼) is called the filter associated with the ideal 𝐼. A non-trivial ideal 𝐼 ⊂ 𝑃(ℕ) is called an admissible ideal in ℕ if and only if it contains all singletons i.e., if it contains {{𝑥}: 𝑥 ∈ ℕ}. An admissible ideal 𝐼 ⊂ 𝑃(ℕ) is said to be satisfy the condition (𝐴𝑃) if for every countable family of mutually disjoint sets {𝐴1 , 𝐴2 . . . } belonging to 𝐼 there exists a countable family {𝐵1 , 𝐵2 . . . } in 𝐼 such that 𝐴𝑖 𝛥𝐵𝑖 is a finite set for each 𝑖 ∈ ℕ and 𝐵 =∪∞ 𝑖=1 𝐵𝑖 ∈ 𝐼. Definition 1.9 A sequence 𝑥 = (𝑥𝑘 ) of numbers is said to be 𝐼 −convergent to a number 𝐿 provided that, for every 𝜀 > 0, 𝑘 ∈ ℕ: |𝑥𝑘 − 𝐿| ≥ 𝜀 ∈ 𝐼 Further, Das 𝑒𝑡 𝑎𝑙. [6] extended the notion of 𝐼 −convergence to 𝐼 −statistical convergence and 𝐼 −lacunary statistical convergence as follows. Definition 1.10 A sequence 𝑥 = (𝑥𝑘 ) of numbers is said to be 𝐼 −statistically convergent to 𝐿 provided that, for every 𝜀 > 0 and each 𝛾 > 0, 𝑛 ∈ ℕ:

1 {𝑘 ≤ 𝑛: |𝑥𝑘 − 𝐿| ≥ 𝜀} ≥ 𝛾 ∈ 𝐼 𝑛

Moreover, above definition is also generalized for lacunary sequences as follows. Definition 1.11 Let 𝜃 = (𝑘𝑟 ) be a lacunary sequence as defined above. A sequence 𝑥 = (𝑥𝑘 ) of numbers is said to be 𝐼 −lacunary statistically convergent to 𝐿 or 𝑆𝜃 (𝐼) −convergent to 𝐿 provided that, for every 𝜀 > 0 and each 𝛾 > 0, 𝑟 ∈ ℕ:

1 {𝑘 ∈ 𝐼𝑟 : |𝑥𝑘 − 𝐿| ≥ 𝜀} ≥ 𝛾 ∈ 𝐼. ℎ𝑟

In this case we write 𝑥𝑘 → 𝐿(𝑆𝜃 (𝐼)). Definition1.12 A sequence 𝑥 = (𝑥𝑘 ) of numbers is said to be 𝑁𝜃 (𝐼) − convergent to 𝐿 provided that, for every 𝜀 > 0,

𝑟 ∈ ℕ:

1 {𝑘 ∈ 𝐼𝑟 : ℎ𝑟

‍|𝑥𝑘 − 𝐿)| ≥ 𝜀}

∈ 𝐼.

𝑘∈𝐼𝑟

In this case we write 𝑥𝑘 → 𝐿(𝑁𝜃 (𝐼)). Some further applications of 𝐼 −convergence can be found in [13] ,[14] and [19]. Throughout this paper, 𝑋 will represent a Banach space, 𝜃 = (𝑘𝑟 ) be a lacunary sequence as defined above, 𝑥 = (𝑥𝑘 ) be a sequence of elements of 𝑋 and 𝐼 ⊆ 𝑃(ℕ) be a non trivial admissible ideal (as otherwise indicated). 26

2. Weak 𝑰 − lacunary statistical convergence Nuray [21], defined weak lacunary statistical convergence and proved some relations between weak lacunary statistical convergence and weak statistical convergence. In this section, we generalized these notions and study relevent connections with the help of ideals. Definition 2.1 A sequence (𝑥𝑘 ) is said to be weak statistically convergent to 𝑥 with respect to 𝐼 in 𝑋 if for every 𝜀 > 0, every 𝛾 > 0 and each 𝑓 ∈ 𝑋 ∗ , where 𝑋 ∗ is the continuous dual of 𝑋 we have 1 𝑛 ∈ ℕ: |{𝑘 ≤ 𝑛: |𝑓(𝑥𝑘 − 𝑥)| ≥ 𝜀}| ≥ 𝛾 ∈ 𝐼. 𝑛 In this case we write 𝑊𝑆(𝐼) − lim𝑥𝑘 = 𝑥. We will denote the set of all weak 𝐼 − statistically convergent sequences by 𝑊𝑆(𝐼). For 𝐼 = 𝐼𝑓𝑖𝑛 , weak 𝐼 −statistical convergence coincides with weak statistical convergence of [2]. Definition 2.2 A sequence (𝑥𝑘 ) is said to be weak lacunary statistically convergent to 𝑥 with respect to 𝐼 in 𝑋 if for every 𝜀 > 0, every 𝛾 > 0 and each 𝑓 ∈ 𝑋 ∗ , where 𝑋 ∗ is the continuous dual of 𝑋 we have 𝑟 ∈ ℕ:

1 |{𝑘 ∈ 𝐼𝑟 : |𝑓(𝑥𝑘 − 𝑥)| ≥ 𝜀}| ≥ 𝛾 ∈ 𝐼. ℎ𝑟

In this case we write 𝑊𝑆𝜃 (𝐼) − lim𝑥𝑘 = 𝑥. We will denote the set of all weak 𝐼 − lacunary statistically convergent sequences by 𝑊𝑆𝜃 (𝐼). For 𝐼 = 𝐼𝑓𝑖𝑛 , weak 𝐼 −lacunary statistical convergence coincides with weak lacunary statistical convergence of [21]. Definition 2.3 A sequence (𝑥𝑘 ) is said to be weak 𝑁𝜃 −convergent to 𝑥 with respect to 𝐼 in 𝑋 if for every 𝜀 > 0 and each 𝑓 ∈ 𝑋 ∗ , 𝑟 ∈ ℕ:

1 ℎ𝑟

‍|𝑓(𝑥𝑘 − 𝑥)| ≥ 𝜀 ∈ 𝐼. 𝑘∈𝐼𝑟

In this case we write 𝑊𝑁𝜃 (𝐼) − lim𝑥𝑘 = 𝑥. We will denote the set of all weak 𝑁𝜃 (𝐼) −convergent sequences by 𝑊𝑁𝜃 (𝐼). For 𝐼 = 𝐼𝑓𝑖𝑛 , weak 𝑁𝜃 (𝐼) −convergence coincides with weak 𝑁𝜃 −convergence of [21]. Theorem 2.1 Let 𝐼 ⊂ 𝑃(ℕ) be an admissible ideal. If 𝑊𝑆𝜃 − lim𝑘 𝑥𝑘 = 𝑥, then 𝑊𝑆𝜃 (𝐼) − lim𝑘 𝑥𝑘 = 𝑥. Proof Suppose 𝑊𝑆𝜃 − lim𝑥𝑘 = 𝑥, then for each 𝜀 > 0, each 𝑓 ∈ 𝑋 ∗ , lim

𝑟→∞

1 𝑘 ∈ 𝐼𝑟 : 𝑓 𝑥𝑘 − 𝑥 ℎ𝑟

≥ 𝜀 = 0.

This immediately follows that for each 𝛾 > 0, there exists a positive integer 𝑚 such that 1 𝑘 ∈ 𝐼𝑟 : 𝑓 𝑥𝑘 − 𝑥 ℎ𝑟

≥ 𝜀 < 𝛾 for all 𝑛 ≥ 𝑚

𝑖. 𝑒. 𝑟 ∈ ℕ:

1 |{𝑘 ∈ 𝐼𝑟 : |𝑓(𝑥𝑘 − 𝑥)| ≥ 𝜀}| ≥ 𝛾 ⊆ {1,2,3, . . . . , 𝑚 − 1} ℎ𝑟

Since 𝐼 is admissible, therefore 𝑟 ∈ ℕ:

1 |{𝑘 ∈ 𝐼𝑟 : |𝑓(𝑥𝑘 − 𝑥)| ≥ 𝜀}| ≥ 𝛾 ∈ 𝐼. ℎ𝑟

This shows that 𝑊𝑆𝜃 (𝐼) − lim𝑥𝑘 = 𝑥. ∎ The converse of above result is not true in general and can be seen from the following example. Example 2.1 Let 𝑋 = 𝑐00 be the normed linear space with ||. ||𝑝 (1 < 𝑝 < ∞), 𝐼 be an admissible ideal, 𝐴 ∈ 𝐼 is fixed and 𝜃 = (2𝑟 ) be a lacunary sequence as defined above. Define a sequence (𝑥𝑘 ) in 𝑐00 by 𝑘, (𝑘) 𝑥𝑗

=

𝑘, 0,

2𝑟−1 + 1 ≤ 𝑘 ≤ 2𝑟−1 + [ ℎ𝑟 ]

𝑖𝑓 𝑗 ≤ 𝑘, 𝑖𝑓 𝑗 ≤ 𝑘, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒,

2

𝑟−1

𝑎𝑛𝑑 𝑟 ∉ 𝐴;

𝑟

≤ 𝑘 ≤ 2 𝑎𝑛𝑑 𝑟 ∈ 𝐴;

where 𝐼𝑟 = (2𝑟−1 , 2𝑟 ]. For arbitrary 𝑓 ∈ 𝑋 ∗ , there is unique 𝑦 ∈ 𝑙𝑞 such that ∞ (𝑘)

𝑓(𝑥𝑘 ) =

‍𝑥𝑗

𝑦𝑗 ≤∥ 𝑥 ∥𝑝 ∥ 𝑦 ∥𝑞

(𝑏𝑦 𝐻𝑜𝑙𝑑𝑒𝑟 𝑖𝑛𝑒𝑞𝑢𝑎𝑙𝑖𝑡𝑦)

(1)

𝑗 =1

So, for each 𝜀 > 0 (0 < 𝜀 < 1) and each 𝜑 ∈ 𝑋 ∗ , 1 𝑘 ∈ 𝐼𝑟 : |𝑓(𝑥𝑘 − 0)| ≥ 𝜀 ℎ𝑟

≤

1 𝑘 ∈ 𝐼𝑟 : ∥ 𝑥 ∥𝑝 ∥ 𝑦 ∥𝑞 ≥ 𝜀 ℎ𝑟 27

=

1 ℎ𝑟

𝜀 ∥ 𝑦 ∥𝑞

𝑘 ∈ 𝐼𝑟 : ∥ 𝑥 ∥𝑝 ≥

=

1 ℎ𝑟

(𝑘)

𝑘 ∈ 𝐼𝑟 : 𝑥𝑗

= 𝑘

=

[ ℎ𝑟 ] → 0 as 𝑟 → ∞ and 𝑟 ∉ 𝐴 ℎ𝑟

Thus, for each 𝛾 > 0,

𝑟 ∈ ℕ:

1 𝑘 ∈ 𝐼𝑟 : |𝑓(𝑥𝑘 − 0)| ≥ 𝜀 ℎ𝑟

≥ 𝛾 ⊂ 𝐴 ∪ {1,2, … , 𝑚}

for some 𝑚 ∈ ℕ. 𝑊𝑆𝜃 (𝐼)

Since 𝐴 ∈ 𝐼 and 𝐼 is admissible, it gives that 𝑥𝑘

→

0. Also by (1) we can see that,

∞ (𝑘)

𝑓(𝑥𝑘 ) =

‍𝑥𝑗

𝑦(𝑗) ≤∥ 𝑥 ∥𝑝 ∥ 𝑦 ∥𝑞

(by Holder inequality)

𝑗 =1

1 𝑝

∞

=

‍

‍ 𝑦(𝑗)

𝑗 =1

1 𝑝

‍

1 𝑞

∞

(𝑘) 𝑝 𝑥𝑗

‍ 𝑦(𝑘)

𝑗 =1

𝑞

(by structure of the sequence)

𝑘=1

1 𝑝

𝑘

‍( 𝑘)𝑝

=

𝑞

𝑗 =1

𝑘

=

1 𝑞

∞

(𝑘) 𝑝 𝑥𝑗

1

1 1 1 + 𝑝 𝑀𝑞 .

𝑀𝑞

(for some positive constant 𝑀) = 𝑘 2

𝑗 =1

This gives that, 1 ℎ𝑟

‍|𝑓(𝑥𝑘 − 0)| ≤ 𝑘∈𝐼𝑟

𝑊𝑁𝜃

1 ℎ𝑟

1 1 1 + 𝑝 𝑀𝑞

‍𝑘 2

↛ 0 𝑎𝑠 𝑟 → ∞,

𝑘∈𝐼𝑟

𝑊𝑆𝜃

𝑖. 𝑒., 𝑥𝑘 ↛ 0. Hence, by Theorem [4 ] of [21], we have 𝑥𝑘 ↛ 0. ∎ Theorem 2.2 For a lacunary sequence 𝜃 = (𝑘𝑟 ), the sequence (𝑥𝑘 ) is weak 𝑁𝜃 (𝐼) −convergent to 𝑥 if and only if (𝑥𝑘 ) is weak 𝐼 −lacunary statistically convergent to 𝑥. Proof Let (𝑥𝑘 ) is weak 𝑁𝜃 (𝐼) −convergent to 𝑥, then for every 𝜀 > 0 and each 𝑓 ∈ 𝑋 ∗ , we have 𝑟 ∈ ℕ:

1 ℎ𝑟

‍|𝑓(𝑥𝑘 − 𝑥)| ≥ 𝜀 ∈ 𝐼. 𝑘∈𝐼𝑟

Also for given 𝜀 > 0, we have 1 ℎ𝑟

‍|𝑓(𝑥𝑘 − 𝑥)| ≥ 𝑘∈𝐼𝑟

≥

1 ℎ𝑟

‍ 𝑓(𝑥𝑘 − 𝑥) 𝑘∈𝐼𝑟 𝑓(𝑥 𝑘 −𝑥) ≥𝜀

𝜀 {𝑘 ∈ 𝐼𝑟 : 𝑓(𝑥𝑘 − 𝑥) ≥ 𝜀} . ℎ𝑟

Hence, for every 𝜀 > 0, each 𝛾 > 0 and each 𝑓 ∈ 𝑋 ∗ we have 𝑟 ∈ ℕ:

1 1 |{𝑘 ∈ 𝐼𝑟 : |𝑓(𝑥𝑘 − 𝑥)| ≥ 𝜀}| ≥ 𝛾 ⊆ 𝑟 ∈ ℕ: ℎ𝑟 ℎ𝑟

‍|𝑓(𝑥𝑘 − 𝑥)| ≥ 𝜀𝛾 ∈ 𝐼. 𝑘∈𝐼𝑟

This immediately gives (𝑥𝑘 ) is weak 𝐼 −lacunary statistically convergent to 𝑥. Conversely, suppose that (𝑥𝑘 ) is weak 𝐼 −lacunary statistically convergent to 𝑥. Then for every 𝜀 > 0, every 𝛾 > 0 and each 𝑓 ∈ 𝑋 ∗ we have

𝑟 ∈ ℕ:

1 |{𝑘 ∈ 𝐼𝑟 : |𝑓(𝑥𝑘 − 𝑥)| ≥ 𝜀}| ≥ 𝛾 ∈ 𝐼. ℎ𝑟

Since 𝑓 ∈ 𝑋 ∗ , therefore 𝑓 is bounded. Let |𝑓(𝑥𝑘 − 𝑥)| ≤ 𝑀 for all 𝑘. For given 𝜀 > 0, we have 28

1 ℎ𝑟

‍|𝑓(𝑥𝑘 − 𝑥)| = 𝑘∈𝐼𝑟

≤

1 ℎ𝑟

‍ 𝑓(𝑥𝑘 − 𝑥) + 𝑘∈𝐼𝑟 𝑓(𝑥 𝑘 −𝑥) ≥𝜀

‍ 𝑓(𝑥𝑘 − 𝑥) 𝑘∈𝐼𝑟 𝑓(𝑥 𝑘 −𝑥) 1. Proof Let liminf𝑟 𝑞𝑟 > 1, then there exists a 𝛿 > 0 such that 𝑞𝑟 > 1 + 𝛿 for sufficiently large 𝑟 which implies that ∗

𝑘𝑟 ℎ𝑟

≤

𝛿+1 𝛿

. Let 𝑊𝑆(𝐼) −

lim𝑥𝑘 = 𝑥, then for every 𝜀 > 0, every 𝛾 > 0 and each 𝑓 ∈ 𝑋 , we have 1 𝑛 ∈ ℕ: |{𝑘 ≤ 𝑛: |𝑓(𝑥𝑘 − 𝑥)| ≥ 𝜀}| ≥ 𝛾 ∈ 𝐼. 𝑛 Also, one can easily observe that 1 1 {𝑘 ≤ 𝑘𝑟 : |𝑓(𝑥𝑘 − 𝑥)| ≥ 𝜀} ≥ {𝑘 ∈ 𝐼𝑟 : |𝑓(𝑥𝑘 − 𝑥)| ≥ 𝜀} 𝑘𝑟 𝑘𝑟 ℎ𝑟 1 {𝑘 ∈ 𝐼𝑟 : |𝑓(𝑥𝑘 − 𝑥)| ≥ 𝜀} 𝑘𝑟 ℎ𝑟

= ≥

𝛿

1

𝛿+1 ℎ 𝑟

{𝑘 ∈ 𝐼𝑟 : |𝑓(𝑥𝑘 − 𝑥)| ≥ 𝜀} ;

So, 𝑟 ∈ ℕ:

1 𝑘 ∈ 𝐼𝑟 : 𝑓 𝑥𝑘 − 𝑥 ℎ𝑟

≥𝜀

≥ 𝛾 ⊆ 𝑟 ∈ ℕ:

1 𝛾𝛿 {𝑘 ≤ 𝑘𝑟 : |𝑓(𝑥𝑘 − 𝑥)| ≥ 𝜀} ≥ ∈ 𝐼, 𝑘𝑟 𝛿+1

which gives the result. Conversely, suppose that liminf𝑟 𝑞𝑟 = 1, then following as in [11], one can find a subsequence (𝑘𝑟(𝑗 ) ) of 𝜃 = (𝑘𝑟 ) such that 𝑘𝑟(𝑗 ) 1 < 1+ 𝑘𝑟(𝑗 )−1 𝑗

𝑎𝑛𝑑

𝑘𝑟(𝑗 )−1 >𝑗 𝑘𝑟(𝑗 −1)

𝑤ℎ𝑒𝑟𝑒

𝑟(𝑗) ≥ 𝑟(𝑗 − 1) + 2

Define a sequence (𝑥𝑘 ) as follows

𝑥𝑘 =

1, 0,

𝑖𝑓 𝑘 ∈ 𝐼𝑟 (𝑗 ) , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 .

Since 𝑓 ∈ 𝑋 ∗ , therefore 𝑓 is bounded. Let |𝑓(𝑥𝑘 − 𝑥)| ≤ 𝑀 for all 𝑘. Thus for any 𝑓 ∈ 𝑋 ∗ , 𝑥 ∈ 𝑋, 1 ℎ𝑟(𝑗 )

‍|𝑓(𝑥𝑘 − 𝑥)| = 𝑘∈𝐼𝑟(𝑗 )

1 ℎ𝑟(𝑗 )

‍|𝑓(1 − 𝑥)| ≤ 𝑀 𝑓𝑜𝑟 𝑗 = 1,2,⋅⋅⋅ 𝑘∈𝐼𝑟(𝑗 )

and 1 ℎ𝑟

‍|𝑓(𝑥𝑘 − 𝑥)| = 𝑘∈𝐼𝑟

1 ℎ𝑟

‍|𝑓(−𝑥)| ≤ 𝑀 𝑓𝑜𝑟 𝑟 ≠ 𝑟𝑗 𝑘∈𝐼𝑟

which gives (𝑥𝑘 ) ∉ 𝑊𝑁𝜃 (𝐼). By Theorem 2.2 we have (𝑥𝑘 ) ∉ 𝑊𝑆𝜃 (𝐼). Let 𝑘𝑟(𝑗 −1) ≤ 𝑛 ≤ 𝑘𝑟(𝑗 ) , then 1 𝑛

𝑛

𝑘=1

1 ‍|𝑓(𝑥𝑘 − 𝑥)| ≥ 𝑛

≥

𝑛

‍ 𝑓(𝑥𝑘 − 𝑥) 𝑘=1 𝑓(𝑥 𝑘 −𝑥) ≥𝜀

𝜀 {𝑘 ≤ 𝑛: 𝑓(𝑥𝑘 − 𝑥) ≥ 𝜀} . 𝑛

We can write, 29

𝜀 1 {𝑘 ≤ 𝑛: 𝑓(𝑥𝑘 − 𝑥) ≥ 𝜀} ≤ 𝑛 𝑛

𝑛

𝑘=1

1 1 2𝑀 ‍|𝑓(𝑥𝑘 − 𝑥)| ≤ 𝑀( + ) = 𝑗 𝑗 𝑗

Thus, (𝑥𝑘 ) is weak 𝐼 − statistically convergent to 𝑥. ∎ In [21] it has been shown that for any lacunary sequence 𝜃 = (𝑘𝑟 ), 𝑊𝑆𝜃 ⊆ 𝑊𝑆 if and only if limsup𝑟 𝑞𝑟 < ∞. However, for any admissible ideal 𝐼 this is not sure in general. Theorem 2.4 Let 𝐼 be an admissible ideal having condition (AP) and 𝜃 ∈ 𝐹(𝐼). If 𝑥 ∈ 𝑊𝑆(𝐼) ∩ 𝑊𝑆𝜃 (𝐼), then 𝑊𝑆(𝐼) − lim𝑥𝑘 = 𝑊𝑆𝜃 (𝐼) − lim𝑥𝑘 . Proof Suppose 𝑊𝑆(𝐼) − lim𝑥𝑘 = 𝑥 and 𝑊𝑆𝜃 (𝐼) − lim𝑥𝑘 = 𝑦 and 𝑥 ≠ 𝑦. Let |𝑥 − 𝑦| > 2𝜀, (𝜀 > 0). Since 𝐼 satisfy the condition (AP) so there exists 𝑀 ∈ 𝐹(𝐼) such that lim

𝑟→∞

1 {𝑘 ≤ 𝑚𝑟 : 𝑓(𝑥𝑘 − 𝑥) ≥ 𝜀} = 0, 𝑚𝑟

where 𝑀 = 𝑚1 , 𝑚2 , 𝑚3 . . . . . Let, 𝐺 = 𝑘 ≤ 𝑚𝑟 : 𝑓 𝑥𝑘 − 𝑥

≥ 𝜀 and

𝐻 = 𝑘 ≤ 𝑚𝑟 : 𝑓 𝑥𝑘 − 𝑦 Since lim𝑟→∞

|𝐺| 𝑚𝑟

= 0, it follows that lim𝑟→∞

1 𝑚𝑟

≥𝜀 .

{𝑘 ≤ 𝑚𝑟 : 𝑓(𝑥𝑘 − 𝑦) ≥ 𝜀} = 1.

𝑡ℎ Let, 𝑀 ∗ = 𝑘𝑙1 , 𝑘𝑙2 , 𝑘𝑙3 , … = 𝑀 ∩ 𝜃 ∈ 𝐹(𝐼). Then the 𝑘𝑙𝑝 term of the weak statistical limit expression

1 1 {𝑘 ≤ 𝑚𝑟 : 𝑓(𝑥𝑘 − 𝑦) ≥ 𝜀} : 𝑚𝑟 𝑘𝑙 𝑝

𝑙𝑝

𝑘∈

1 = 𝑘𝑙 𝑝

‍𝐼𝑟 : 𝑓(𝑥𝑘 − 𝑦) ≥ 𝜀 𝑟=1

= where 𝑡𝑟 =

1 ℎ𝑟

𝑙𝑝

‍ {𝑘 ∈ 𝐼𝑟 : 𝑓(𝑥𝑘 − 𝑦) ≥ 𝜀} 𝑟 =1

1

𝑙𝑝

‍ℎ𝑟 𝑡𝑟 𝑙𝑝 ‍ ℎ 𝑟 𝑟=1 𝑟=1

(2)

{𝑘 ∈ 𝐼𝑟 : 𝑓(𝑥𝑘 − 𝑦) ≥ 𝜀} and 𝑡𝑟 ⟶𝐼 0 as 𝑊𝑆𝜃 (𝐼) − lim𝑥𝑘 = 𝑦. Since 𝜃 is a lacunary sequence and the right side of above

expression is a regular weighted mean transform of the sequence 𝑡 = 𝑡𝑟 , therefore it is also 𝐼-convergent to zero as 𝑝 → ∞. Also 𝐼 satisfies 1 the condition (AP) which gives it has a subsequence converging to zero. Since this is a subsequence of {𝑘 ≤ 𝑚𝑟 : 𝑓(𝑥𝑘 − 𝑦) ≥ 𝜀} , we infer that

1 𝑚𝑟

𝑚𝑟

{𝑘 ≤ 𝑚𝑟 : 𝑓(𝑥𝑘 − 𝑦) ≥ 𝜀} is not convergent to 1 which is a contradiction. This completes the proof of the theorem. ∎

3. Weak 𝑰 −lacunary statistical limit and cluster point Fridy [10] introduced the notion of statistical limit point and statistical cluster point for the sequences of numbers. Connor 𝑒𝑡 𝑎𝑙. [4] generalised these notions to 𝐴 −statistical limit point and 𝐴 −statistical cluster point where 𝐴 is a nonnegative regular summability matrix. Demirci [7] generalised these notions to 𝜃 −statistical limit point and 𝜃 −statistical cluster point for sequences of numbers. Gürdal 𝑒𝑡 𝑎𝑙. [14] have generalized the notion of 𝐴 −statistical limit point and 𝐴 −statistical cluster point to define 𝐴𝐼 −statistical limit point and 𝐴𝐼 −statistical cluster point where 𝐼 ⊂ 𝑃(ℕ) is an ideal. Further these concepts has been defined in different spaces. Now we will introduce the notion of weak 𝐼 −lacunary statistical limit and cluster point in Banach spaces. Let 𝜃 = 𝑘𝑟 be a lacunary sequence. For a Banach space 𝑋, let 𝑥 = 𝑥𝑘 be a sequence in 𝑋. Let 𝑥𝑘 𝐾 = 𝑘 𝑗 : 𝑗 ∈ ℕ , then we denote 𝑥𝑘 𝑗 by (𝑥)𝐾 . If for every 𝛾 > 0, 𝑟 ∈ ℕ:

𝑗

be a subsequence of 𝑥 and

1 𝑘 𝑗 ∈ 𝐼𝑟 : 𝑗 ∈ ℕ ≥ 𝛾 ∈ 𝐼; ℎ𝑟

then (𝑥)𝐾 is called 𝜃 𝐼 -thin subsequence. on the other hand (𝑥)𝐾 is a 𝜃 𝐼 -nonthin subsequence of 𝑥 provided that 𝑟 ∈ ℕ:

1 𝑘 𝑗 ∈ 𝐼𝑟 : 𝑗 ∈ ℕ ≥ 𝛾 ∉ 𝐼. ℎ𝑟

Definition 3.1 Let 𝑋 be a Banach space, 𝜃 = (𝑘𝑟 ) be a lacunary sequence and 𝑓 ∈ 𝑋 ∗ . An element 𝑥 ∈ 𝑋 is called a weak 𝑆𝜃 (𝐼) −limit point of a sequence 𝑥 = (𝑥𝑘 ) in 𝑋 provided that there is a 𝜃(𝐼)-nonthin subsequence of (𝑥𝑘 ) that is weak convergent to 𝑥. Let Λ(𝑊𝑆𝜃 (𝐼), 𝑥) denotes the set of all 𝑊𝑆𝜃 (𝐼)-limit points of the sequence 𝑥 = (𝑥𝑘 ). Definition 3.2 Let 𝑋 be a Banach space, 𝜃 = 𝑘𝑟 be a lacunary sequence and 𝑓 ∈ 𝑋 ∗ . An element 𝑥 ∈ 𝑋 is called a weak 𝑆𝜃 𝐼 −cluster point of a sequence 𝑥 = 𝑥𝑘 in 𝑋 provided that for every 𝜀 > 0 and each 𝑓 ∈ 𝑋 ∗ , we have 𝑟 ∈ ℕ:

1 𝑘 ∈ 𝐼𝑟 : 𝑓 𝑥𝑘 − 𝑥 ℎ𝑟

0, 𝑟 ∈ ℕ:

1 {𝑘 𝑗 ∈ 𝐼𝑟 : 𝑗 ∈ ℕ} ≥ 𝛾 ∉ 𝐼 ℎ𝑟

(3)

Now, the containment 𝑘 ∈ 𝐼𝑟 : 𝑓(𝑥𝑘 − 𝑥) < 𝜀 ⊇ {𝑘(𝑗) ∈ 𝐼𝑟 : 𝑓(𝑥𝑘(𝑗 ) − 𝑥) < 𝜀} gives 𝑘 ∈ 𝐼𝑟 : 𝑓(𝑥𝑘 − 𝑥) < 𝜀 ⊇ 𝑘 𝑗 ∈ 𝐼𝑟 : 𝑗 ∈ ℕ − 𝑘 𝑗 ∈ 𝐼𝑟 : 𝑓(𝑥𝑘 (𝑗 ) − 𝑥) ≥ 𝜀 ; which immediately implies 1 𝑘 ∈ 𝐼𝑟 : 𝑓(𝑥𝑘 − 𝑥) < 𝜀 ℎ𝑟

≥

1 1 𝑘 𝑗 ∈ 𝐼𝑟 : 𝑗 ∈ ℕ − 𝑘 𝑗 ∈ 𝐼𝑟 : 𝑓(𝑥𝑘(𝑗 ) − 𝑥) ≥ 𝜀 . ℎ𝑟 ℎ𝑟

So, for each 𝛾 > 0 and 𝜀 > 0, 𝑟 ∈ ℕ:

1 𝑘 ∈ 𝐼𝑟 : 𝑓(𝑥𝑘 − 𝑥) < 𝜀 ℎ𝑟

≥ 𝛾 ⊇ 𝑟 ∈ ℕ: 𝑘 𝑗 ∈ 𝐼𝑟 : 𝑗 ∈ ℕ ≥ 𝛾 − 𝑟 ∈ ℕ: 𝑘 𝑗 ∈ 𝐼𝑟 : 𝑓(𝑥𝑘 (𝑗 ) − 𝑥) ≥ 𝜀

= 𝑟 ∈ ℕ: 𝑘 𝑗 ∈ 𝐼𝑟 : 𝑗 ∈ ℕ ≥ 𝛾 ∩ 𝑟 ∈ ℕ: 𝑘 𝑗 ∈ 𝐼𝑟 : 𝑓(𝑥𝑘(𝑗 ) − 𝑥) ≥ 𝜀

≥𝛾

= 𝑟 ∈ ℕ: 𝑘 𝑗 ∈ 𝐼𝑟 : 𝑗 ∈ ℕ ≥ 𝛾 ∩ 𝑟 ∈ ℕ: 𝑘 𝑗 ∈ 𝐼𝑟 : 𝑓(𝑥𝑘 (𝑗 ) − 𝑥) ≥ 𝜀

0 and each 𝑓 ∈ 𝑋 ∗ , the set 𝑘 𝑗 ∈ 𝐼𝑟 : 𝑓(𝑥𝑘 (𝑗 ) − 𝑥) ≥ 𝜀 is finite for which we have lim𝑟→∞

1 𝑘 𝑗 ∈ 𝐼𝑟 : 𝑓(𝑥𝑘(𝑗 ) − 𝑥) ≥ 𝜀 ℎ𝑟

=0,

𝑖. 𝑒. 𝑊𝑆𝜃 − lim𝑥𝑘 (𝑗 ) = 𝑥. Using Theorem 2.1, we get 𝑊𝑆𝜃 (𝐼) − lim𝑥𝑘(𝑗 ) = 𝑥. Thus, for each 𝜀 > 0, each 𝛾 > 0 and each 𝑓 ∈ 𝑋 ∗ 𝑟 ∈ ℕ:

1 |𝑘 ∈ 𝐼𝑟 : |𝑓(𝑥𝑘 − 𝑥)| ≥ 𝜀| ≥ 𝛾 ∈ 𝐼 ℎ𝑟

or

𝑟 ∈ ℕ:

1 |𝑘 ∈ 𝐼𝑟 : |𝑓(𝑥𝑘 − 𝑥)| ≥ 𝜀| < 𝛾 ∈ 𝐹 𝐼 ℎ𝑟

(5)

Using (3) and (5) in (4), we get 𝑟 ∈ ℕ:

1 ℎ𝑟

𝑘 ∈ 𝐼𝑟 : 𝑓(𝑥𝑘 − 𝑥) < 𝜀

≥ 𝛾 ∈ 𝐹(𝐼).

𝑖. 𝑒. 𝑟 ∈ ℕ:

1 𝑘 ∈ 𝐼𝑟 : 𝑓(𝑥𝑘 − 𝑥) < 𝜀 ℎ𝑟

≥ 𝛾 ∉ 𝐼.

This shows that 𝑥 ∈ Γ(𝑊𝑆𝜃 (𝐼), 𝑥) and therefore we have the containment Λ(𝑊𝑆𝜃 (𝐼), 𝑥) ⊆ Γ(𝑊𝑆𝜃 (𝐼), 𝑥). ∎ Theorem 3.2 Let 𝜃 = 𝑘𝑟 be a lacunary sequence. For any sequence 𝑥 = (𝑥𝑘 ) in a Banach space 𝑋, if 𝑊𝑆𝜃 (𝐼) − lim𝑘 𝑥𝑘 = 𝑥0 , then Λ 𝑊𝑆𝜃 (𝐼), 𝑥 = Γ 𝑊𝑆𝜃 (𝐼), 𝑥 = {𝑥0 }. Proof. We first show that Λ 𝑊𝑆𝜃 (𝐼) , 𝑥 = 𝑥0 . Let Λ 𝑊𝑆𝜃 (𝐼), 𝑥 = 𝑥0 , 𝑦0 such that 𝑥0 ≠ 𝑦0 . By definition there exist two 𝜃(𝐼)-nonthin subsequences 𝑥𝑘(𝑖) and 𝑥𝑙(𝑗 ) of the sequence 𝑥 = (𝑥𝑘 ) which are respectively weakly convergent to 𝑥0 and 𝑦0 . Choose 𝜀 > 0 such that |𝑓(𝑥 )−𝑓(𝑦 )|

0 0 0 0 and 𝛾 > 0, the set 𝑟 ∈ ℕ: = 𝑟 ∈ ℕ:

1 𝑙 𝑗 ∈ 𝐼𝑟 : 𝑓(𝑥𝑙(𝑗 ) − 𝑦0 ) < 𝜀 ℎ𝑟

≥𝛾

1 1 𝑙 𝑗 ∈ 𝐼𝑟 : 𝑗 ∈ ℕ ≥ 𝛾 − 𝑟 ∈ ℕ: 𝑙 𝑗 ∈ 𝐼𝑟 : 𝑓(𝑥𝑙(𝑗 ) − 𝑦0 ) ≥ 𝜀 ℎ𝑟 ℎ𝑟

≥𝛾 31

= 𝑟 ∈ ℕ:

1 1 𝑙 𝑗 ∈ 𝐼𝑟 : 𝑗 ∈ ℕ ≥ 𝛾 ∩ 𝑟 ∈ ℕ: 𝑙 𝑗 ∈ 𝐼𝑟 : 𝑓(𝑥𝑙(𝑗 ) − 𝑦0 ) ≥ 𝜀 ℎ𝑟 ℎ𝑟

𝑐

≥𝛾 .

Since 𝑙(𝑗) is 𝜃(𝐼) −nonthin subsequence and 𝑊𝑆𝜃 (𝐼) − lim𝑥𝑙(𝑗 ) = 𝑦0 , therefore 𝑟 ∈ ℕ:

1 1 𝑙 𝑗 ∈ 𝐼𝑟 : 𝑗 ∈ ℕ ≥ 𝛾 ∩ 𝑟 ∈ ℕ: 𝑙 𝑗 ∈ 𝐼𝑟 : 𝑓(𝑥𝑙(𝑗 ) − 𝑦0 ) ≥ 𝜀 ℎ𝑟 ℎ𝑟

𝑐

≥𝛾

∈ 𝐹(𝐼).

This gives, 𝑟 ∈ ℕ:

1 𝑙 𝑗 ∈ 𝐼𝑟 : 𝑓(𝑥𝑙(𝑗 ) − 𝑦0 ) < 𝜀 ℎ𝑟

≥𝛾 ∈𝐹 𝐼

(6)

Also 𝑊𝑆𝜃 (𝐼) − lim𝑘 𝑥𝑘 = 𝑥0 , so we have 𝑟 ∈ ℕ:

1 𝑙 𝑗 ∈ 𝐼𝑟 : 𝑓(𝑥𝑘 − 𝑥0 ) ≥ 𝜀 ℎ𝑟

≥𝛾 ∈𝐼

(7)

For 0 < 2𝜀 < 𝑓 𝑥0 − 𝑦0 , 𝑙 𝑗 ∈ 𝐼𝑟 : 𝑓(𝑥𝑙(𝑗 ) − 𝑦0 ) < 𝜀 ∩ 𝑘 ∈ 𝐼𝑟 : 𝑓(𝑥𝑘 − 𝑥0 ) < 𝜀 = ∅ so we have, 𝑙 𝑗 ∈ 𝐼𝑟 : 𝑓(𝑥𝑙(𝑗 ) − 𝑦0 ) < 𝜀 ⊆ {𝑘 ∈ 𝐼𝑟 : 𝑓(𝑥𝑘 − 𝑥0 ) ≥ 𝜀}; which implies immadetaly that 1 𝑙 𝑗 ∈ 𝐼𝑟 : 𝑓(𝑥𝑙(𝑗 ) − 𝑦0 ) < 𝜀 ℎ𝑟

≤

1 {𝑘 ∈ 𝐼𝑟 : 𝑓(𝑥𝑘 − 𝑥0 ) ≥ 𝜀} . ℎ𝑟

For 𝜀 > 0, 𝛾 > 0 𝑟 ∈ ℕ:

1 𝑙 𝑗 ∈ 𝐼𝑟 : 𝑓(𝑥𝑙(𝑗 ) − 𝑦0 ) < 𝜀 ℎ𝑟

≥ 𝛾 ⊆ 𝑟 ∈ ℕ:

1 𝑘 ∈ 𝐼𝑟 : 𝑓(𝑥𝑘 − 𝑥0 ) ≥ 𝜀} ≥ 𝛾 ℎ𝑟

Using(7), we have 𝑟 ∈ ℕ:

1 𝑙 𝑗 ∈ 𝐼𝑟 : 𝑓(𝑥𝑙(𝑗 ) − 𝑦0 ) < 𝜀 ℎ𝑟

≥ 𝛾 ∈ 𝐼;

which contradict(6). Hence, Λ 𝑊𝑆𝜃 (𝐼), 𝑥 = 𝑥0 . Similarly, we can show that Γ 𝑊𝑆𝜃 (𝐼), 𝑥 = 𝑥0 . ∎ Theorem 3.3 If 𝑥 = 𝑥𝑘

and 𝑦 = (𝑦𝑘 )

are two sequences in 𝑋 such that lim𝑟

1 ℎ𝑟

𝑘 ∈ 𝐼𝑟 : 𝑥𝑘 ≠ 𝑦𝑘

= 0, then Λ 𝑊𝑆𝜃 (𝐼), 𝑥 =

Λ 𝑊𝑆𝜃 (𝐼), 𝑦 and Γ 𝑊𝑆𝜃 (𝐼), 𝑥 = Γ 𝑊𝑆𝜃 (𝐼), 𝑦 . Proof Suppose 𝑥 ′ ∈ Λ 𝑊𝑆𝜃 (𝐼), 𝑥 , then there exists a 𝜃(𝐼) −nonthin subsequence (𝑥)𝐾 of the sequence 𝑥 = (𝑥𝑘 ) that weakly converges to 𝑥 ′ . 𝑖. 𝑒. If 𝐾 = 𝑘 𝑗 : 𝑗 ∈ ℕ , then for every 𝛾 > 0, 𝑟 ∈ ℕ: Since, lim𝑟

1 ℎ𝑟

1 {𝑘 𝑗 ∈ 𝐼𝑟 : 𝑗 ∈ ℕ} ≥ 𝛾 ∉ 𝐼 𝑎𝑛𝑑 lim𝑓(𝑥𝑘(𝑗 ) − 𝑥 ′ ) = 0 𝑘 ℎ𝑟

{𝑘 ∈ 𝐼𝑟 : 𝑘 ∈ 𝐾, 𝑥𝑘 ≠ 𝑦𝑘 } = 0, it follows that limsup 𝑟

1 {𝑘 ∈ 𝐼𝑟 : 𝑘 ∈ 𝐾, 𝑥𝑘 = 𝑦𝑘 } > 0 ℎ𝑟

(8)

Therefore, there exists a 𝜃(𝐼) −nonthin subsequence (𝑦)𝐾 of the sequence 𝑦 = (𝑦𝑘 ) that weakly converges to 𝑥 ′ . This shows that 𝑥 ′ ∈ Λ 𝑊𝑆𝜃 (𝐼), 𝑦 and therefore Λ 𝑊𝑆𝜃 (𝐼), 𝑥 ⊆ Λ 𝑊𝑆𝜃 (𝐼), 𝑦 . By symmetry we have Λ 𝑊𝑆𝜃 (𝐼), 𝑦 ⊆ Λ 𝑊𝑆𝜃 (𝐼), 𝑥 . Hence, Λ 𝑊𝑆𝜃 (𝐼), 𝑥 = Λ 𝑊𝑆𝜃 (𝐼), 𝑦 . Similarly we can prove Γ 𝑊𝑆𝜃 (𝐼), 𝑥 = Γ 𝑊𝑆𝜃 (𝐼), 𝑦 . ∎

Acknowledgement The authors are thankful to the reviewers for their valuable comments and suggestions.

References [1]

Bala I., On weak* statistical convergence of sequences of functionals, International J. Pure Appl. Math., 70 (2011), pp. 647-653.

[2]

Bhardwaj V. K., Bala I., On weak statistical convergence, International J. Math. Math. Sc. (2007), Article ID 38530, 9 pages, doi:10.1155/2007/38530.

32

[3]

Connor J. S., The statistical and strong p-Cesàro convergence of sequences, Analysis, 8 (1988), pp. 47-63.

[4]

Connor J. A., Kline J., On statistical limit points and the consistency of statistical convergence, J. Math. Anal. Appl., 197 (1996), pp. 392–399.

[5]

Connor J., Ganichev M. and Kadets V., A characterization of Banach spaces with separable duals via weak statistical convergence, J. Math. Anal. Appl.,244 (2000), pp. 251261.

[6]

Das P., Savas E. and Ghosal K., On generalizations of certain summability methods using ideals, Appl. Math. Letters, 24(2011), pp. 1509-1514.

[7]

Demirci K., On lacunary statistical limit points, Demonstratio Mathematica, 35 (2002), pp. 93-101.

[8]

Fast H., Surla Convergence Statistique, Colloquium Mathematicum, 2 (1951), pp. 241-244.

[9]

Fridy J. A., On statistical convergence, Analysis, 5 (1985), pp. 301-313.

[10] Fridy J. A., Statistical limit points, Proceeding of American Mathematical Society, 118 (1993), pp. 1187-1192. [11] Fridy J. A. and Orhan C., Lacunary statistical convergence, Pacific Journal of Mathematics, 160 (1993), pp. 43-51. [12] Freedman A. R. and Sember J. J., Densities and summability, Pacific Journal of Mathematics, 95(1981), pp. 293–305. [13] Gürdal M., 𝑆ahiner A., Extremal 𝐼 −limit points of double sequences, Appl. Math. E-Notes, 8 (2008), pp. 131–137. [14] Gürdal M., Sari H., Extremal 𝐴 −statistical limit points via ideals, Journal of the Egyptian Mathematical Society, 22(1) (2013) , pp. 55–58. [15] Karakus S., Statistical convergence on probabilistic normed spaces, Math. Commun., 12 (2007), pp. 11-23. [16] Karakus S., Demirci K. and Duman O., Statistical convergence on intuitionistic fuzzy normed spaces, Chaos, Solitons and Fractals, 35 (2008), pp. 763-769. [17] Kolk E., The statistical convergence in Banach spaces, Acta et Comment. Univ. Tartu., 928 (1991), pp. 41-52. [18] Kostyrko P., Šalát T. and Wilczynski W., 𝐼 −Convergence, Real Anal. Exchange, 26(2000), pp. 669–686. [19] Kostyrko P., Macaj M., Šalát T. and Sleziak M., 𝐼 −Convergence and Extremal 𝐼 −Limit Points, Math. Slovaca, 55(2005), pp. 443–464. [20] Maddox I. J., Statistical convergence in a locally convex space, Math. Proc. Cambridge Philos. Soc., 104(1)(1988), pp. 141-145. [21] Nuray F., Lacunary weak statistical convergence, Mathematica Bohemica, 136 (2011), pp. 259-268. [22] Šalát T., On statistically convergent sequences of real numbers, Mathematica Slovaca, 30 (2) (1980), pp. 139-150. [23] Schoenberg I. J., The integrability of certain functions and related summability methods, American Mathematical Monthly, 66 (1951), pp. 361-375. [24] Zygmund A., Trigonometric Series, Cambridge University Press, Cambridge, UK, 1979.

33