Rough I-statistical convergence of sequences

arXiv:1611.03224v2 [math.FA] 27 Dec 2016

ROUGH I-STATISTICAL CONVERGENCE OF SEQUENCES PRASANTA MALIK*, MANOJIT MAITY** AND ARGHA GHOSH*

* Department of Mathematics, The University of Burdwan, Golapbag, Burdwan713104, West Bengal, India. Email: [email protected]. , [email protected] ** 25,Teachers Housing Estate, P.O.- Panchasayar, Kolkata-700094, India. Email: [email protected]

Abstract. The concept of I-statistical convergence of a sequence was first introduced by Das et. al. [2]. In this paper we introduce and study the notion of rough I-statistical convergence of sequences in normed linear spaces. We also define the set of all rough I-statistical limits of a sequence and discuss some topological properties of this set.

Key words and phrases : I-statistical convergence, rough I-statistical convergence, I-statistical limit set, I - statistical boundedness. AMS subject classification (2010) : 40A05, 40G99 .

1. Introduction: The notion of convergence of real sequences has been extended to statistical convergence by Fast [7] also independently by Schoenberg [14] using the natural density of N. A subset K of N is said to have natural density d(K) if d(K) = lim

n→∞

|K(n)| n

exists

where K(n) = {j ∈ K : j ≤ n} and |K(n)| represents number of elements in K(n). A sequence x = {xn }n∈N of real numbers is said to be statistically convergent to ζ ∈ R if for any ε > 0, d(A(ε)) = 0, where A(ε) = {n ∈ N : |xn − ζ| ≥ ε}. The study of statistical convergence become one of the most active research area in summability theory after the works of Fridy [6], Salat [15] and many others. The notion of statistical convergence has been further generalized to I-convergence by Kostyrko et. al. [8] using ideals of N. A lot of work on I-convergence can be found in [4, 5] and many others. Recently Das et. al. in [2] introduced the notion of ideal statistical convergence which is a new generalization of the notion 1

2

P. MALIK, M. MAITY AND AR. GHOSH

of statistical convergence. More investigation and application of this notion can be found in [2, 3, 16, 17]. The concept of rough convergence was first introduced by Phu [13]. If x = {xj }j∈N be a sequence in some normed linear space (X, k.k) and r be a non negative real number then x is said to be r-convergent to ξ ∈ X if for any ε > 0, there exists N ∈ N such that k xj − ξ k < r + ε for all j ≥ N . Further this notion of rough convergence has been extended to rough statistical convergence by Aytar [1] using the notion of natural density of N in a similar way as usual notion of convergence was extended to statistical convergence. More work can be found in [9, 10]. Further the notion of rough statistical convergence was generalized to rough I-convergence by Pal et. al. [12] using ideals of N. More investigation and application on this line can be found in [11, 12]. So naturally one can think if the new notion of I-statistical convergence can be introduced in the theory of rough convergence. In this paper we introduce and study the notion of rough I-statistical convergence in a normed linear space (X, k.k) which naturally extends the notions of rough convergence as well as rough statistical convergence in a new way. We also define the set of all rough I-statistical limits of a sequence and proved some topological properties of this set. 2. Basic Definitions and Notations In this section we recall some basic definitions and notations. Definition 2.1. Let X 6= φ. A class I of subsets of X is said to be an ideal in X provided, I satisfies the conditions: (i)φ ∈ I, (ii)A, B ∈ I ⇒ A ∪ B ∈ I, (iii)A ∈ I, B ⊂ A ⇒ B ∈ I. An ideal I in a non-empty set X is called non-trivial if X ∈ / I. Definition 2.2. Let X 6= φ. A non-empty class F of subsets of X is said to be a filter in X provided that: (i)φ ∈ / F, (ii) A, B ∈ F ⇒ A ∩ B ∈ F, (iii)A ∈ F, B ⊃ A ⇒ B ∈ F. Definition 2.3. Let I be a non-trivial ideal in a non-empty set X. Then the class F(I)= {M ⊂ X : ∃A ∈ I such that M = X \ A} is a filter on X. This filter F(I) is called the filter associated with I. A non-trivial ideal I in X(6= φ) is called admissible if {x} ∈ I for each x ∈ X. Throughout the paper we take I as a non-trivial admissible ideal in N unless otherwise mentioned.

ROUGH I-STATISTICAL CONVERGENCE OF SEQUENCES

3

Definition 2.4. [2] Let x = {xk }k∈N be a sequence of real numbers. Then x is said to be I-statistically convergent to ξ if for any ε > 0 and δ > 0 {n ∈ N : n1 |{k ≤ n :k xk − ξ k≥ ε}| ≥ δ} ∈ I. In this case we write I-st- lim x = ξ Definition 2.5. [3] A subset K of N is said to have I-natural density dI (K) if d(K) = I − lim

n→∞

|K(n)| n

exists

where K(n) = {j ∈ K : j ≤ n} and |K(n)| represents the number of elements in K(n). Definition 2.6. [13] If x = {xk }k∈N be a sequence in some normed linear space (X, k.k) and r be a non negative real number then x is said to be r-convergent to ξ ∈ X, if for any ǫ > 0, there exists N ∈ N such that k xk − ξ k< r+ǫ f or all k ≥ N. Definition 2.7. [1] Let x = {xk }k∈N be a sequence in a normed linear space (X, k . k) and r be a non negative real number. x is said to be r- statistically r−st convergent to ξ, denoted by x −→ ξ, if for any ε > 0 we have d(A(ε)) = 0, where A(ε) = {k ∈ N :k xk − ξ k≥ r + ε}. In this case ξ is called the r-statistical limit of x. Definition 2.8. Let x = {xk }k∈N be a sequence in a normed linear space (X, k.k) and r be a non negative real number. Then x is said to be rough Istatistically convergent to ξ or r-I-statistically convergent to ξ if for any ε > 0 and δ > 0 {n ∈ N : n1 |{k ≤ n :k xk − ξ k≥ r + ε}| ≥ δ} ∈ I. In this case ξ is called the rough I-statistical limit of x = {xk }k∈N and we r -I -st denote it by xk −→ ξ. Here r in the above definition is called the roughness degree of the rough Istatistical convergence. If r = 0 we obtain the notion of I-statistical convergence. But our main interest is when r > 0. It may happen that a sequence x = {xk }k∈N is not I-statistically convergent in the usual sense, but there exists a sequence y = {yk }k∈N , which is I-statistically convergent and satisfying the condition kxk − yk k ≤ r for all k ( or for all k whose I-natural density is zero). Then x is rough I-statistically convergent to the same limit. From the above definition it is clear that the rough I-statistical limit of a sequence is not unique. So we consider the set of rough I-statistical limits of a sequence x and we use the notation I-st-LIMrx to denote the set of all rough Istatistical limits of a sequence x. We say that a sequence x is rough I-statistically convergent if I-st-LIMrx 6= φ. Throughout the paper x denotes the sequence {xk }k∈N and X denotes a normed linear space (X, k.k).

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We now provide an example to show that there exists a sequence which is neither rough statistically convergent nor I-statistically convergent but is rough I-statistically convergent. Example 2.1. Let I be an admissible ideal in N. Choose an infinite subset A ∈ I, whose I-natural density is zero but natural density does not exists. We define a sequence x = {xk }k∈N in the following way (−1)k , if k ∈ /A xk = k, if k ∈ A. Then x neither rough statistically convergent nor I-statistically convergent but ∅, if r < 1 I-st-LIMxr = [1 − r, r − 1] , otherwise.

3. Main Results In this section we discuss some basic properties of rough I-statistical convergence of sequences. Theorem 3.1. Let x = {xk }k∈N be a sequence in X and r > 0. Then diam (I-st-LIMrx ) ≤ 2r. In particular if x is I-statistically convergent to ξ, then I-st-LIMrx ⊃ Br (ξ)(= {y ∈ X : ky − ξk ≤ r}) and so diam (I-st-LIMrx ) = 2r. Proof. If possible, let diam (I-st-LIMrx ) > 2r. Then there exist y, z ∈ I-st-LIMrx − r. Let such that ky − zk > 2r. Now choose ε > 0 so that ε < ky−zk 2 A = {k ∈ N : kxk − yk ≥ r + ε} and B = {k ∈ N : kxk − zk ≥ r + ε}. Then 1 n |{k

≤ n : k ∈ A ∪ B}| ≤

1 n |{k

≤ n : k ∈ A}| + n1 |{k ≤ n : k ∈ B}| 1 | {k n→∞ n

and so by the property of I-convergence we have, I- lim I- lim n1 |{k n→∞ 1 |{k ≤ n : n

≤ n : k ∈ A}| +

I- lim n1 |{k n→∞

≤ n : k ∈ A ∪ B} | ≤

≤ n : k ∈ B}| = 0. Thus {n ∈ N :

k ∈ A ∪ B}| ≥ δ} ∈ I for all δ > 0. Let K = {n ∈ N : n1 |{k ≤ n : k ∈ A ∪ B}| ≥ 21 }. Clearly K ∈ I. Now choose n0 ∈ N \ K. Then 1 1 1 / A ∪ B}| ≥ 1 − 21 = 12 n0 |{k ≤ n0 : k ∈ A ∪ B}| < 2 . So n0 |{k ≤ n0 : k ∈ i.e., {k : k ∈ / A ∪ B} is a nonempty set. Take k0 ∈ N such that k0 ∈ / A ∪ B. Then k0 ∈ Ac ∩ B c and hence kxk0 − yk < r + ε and kxk0 − zk < r + ε. So ky − zk ≤ kxk0 − yk + kxk0 − zk ≤ 2(r + ε) < ky − zk, which is absurd. Therefore diam (I-st-LIMrx ) ≤ 2r.


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Now if I-st- lim x = ξ, we proceed as follows. Let ε > 0 and δ > 0 be given. / A we have Then A = {n ∈ N : n1 |{k ≤ n : kxk − ξk ≥ ε}| ≥ δ} ∈ I. Then for n ∈ 1 |{k ≤ n : kx − ξk ≥ ε} < δ, i.e., k n 1 |{k ≤ n : kxk − ξk < ε} ≥ 1 − δ. n Now for each y ∈ Br (ξ) = {y ∈ X : ky − ξk ≤ r} we have

(1)

kxk − yk ≤ kxk − ξk + kξ − yk ≤ kxk − ξk + r.

(2)

Let Bn = {k ≤ n : kxk − ξk < ε}. Then for k ∈ Bn we have kxk − yk < r + ε. Hence Bn ⊂ {k ≤ n : kxk − yk < r + ε}. This implies, |Bnn | ≤ n1 |{k ≤ n : kxk − yk < r + ε}| i.e., n1 |{k ≤ n : kxk − yk < r + ε}| ≥ 1 − δ. Thus for all n ∈ / A, n1 |k ≤ n : kxk − yk ≥ r + ε}| < 1 − (1 − δ). Hence we have {n ∈ N : n1 |{k ≤ n : kxk − yk ≥ r + ε}| ≥ δ} ⊂ A. Since A ∈ I, so {n ∈ N : n1 |{k ≤ n : kxk − yk ≥ r + ε}| ≥ δ} ∈ I. This shows that y ∈ I-st-LIMxr . Therefore I-st-LIMrx ⊃ Br (ξ) and consequently diam(I-st-LIMxr ) ≥ 2r. Hence diam(I-st-LIMxr ) = 2r. In the paper [13] H.X. Phu has already shown that for any subsequence x′ = {xnk }k∈N of a sequence x = {xk }k∈N , LIMxr ⊆ LIMxr′ . But this fact does not hold good in case of rough I-statistical convergence. To support this we cite the following example. Example 3.1. Let I be an admissible ideal of N. Choose an infinite subset A = {j1 < j2 < ....} , A ∈ I, whose I-natural density is zero but natural density does not exist. We define a sequence x = {xn }n∈N in the following way xn Then for r > 0,

=

n, if n = jk f or some jk ∈ A

=

0, otherwise.

I-st-LIMxr

= [−r, r], but I-st-LIMxr′ = ∅ where x′ = {xjn }n∈N .

In the following theorem we show that the rough I-statistical analogue of Phu’s result holds for some kind of subsequences. Theorem 3.2. Let x = {xk }k∈N be a sequence in X and r > 0 be any real number. If x has a subsequence x′ = {xjk } satisfying the condition I- lim n1 |{jk ≤ n→∞

n : k ∈ N}| 6= 0 then I-st-LIMxr ⊂ I-st-LIMxr′ .

Proof. Let ξ ∈ I-st-LIMxr and let x = {xk }k∈N has a subsequence x′ = {xjk }k∈N / such that I- lim n1 |{jk ≤ n : k ∈ N}| 6= 0. Suppose on the contrary, ξ ∈ n→∞

′

1 |{jk n→∞ n

I-st-LIMxr′ . Then there exists an ε > 0 such that I- lim ′

′

≤ n : ||xjk −

≤ n : ||xjk − ξ|| ≥ r + ε }| ≤ n1 |{k ≤ n : ξ|| < r + ε }| 6= 0. Then ′ ′ ||xk − ξ|| ≥ r + ε }| and so I- lim n1 |{k ≤ n : ||xk − ξ|| ≥ r + ε }| 6= 0, which 1 n |{jk

n→∞

6


contradicts that ξ ∈ I-st-LIMxr . Hence ξ ∈ I-st-LIMxr′ . Therefore I-st-LIMxr ⊂ I-st-LIMxr′ . Theorem 3.3. Let x = {xk }k∈N be sequence in X and r > 0 be a real number. Then the rough I-statistical limit set of the sequence x i.e., the set I-st-LIMxr is closed. Proof. If I-st-LIMxr = ∅, then nothing to prove. Let us assume that I-st-LIMxr 6= ∅. Now consider a sequence {yk }k∈N in I-st-LIMxr with lim yk = y. Choose ε > 0 and δ > 0. Then there exists i 2ε ∈ N k→∞

such that kyk − yk < 2ε for all k > i 2ε . Let k0 > i ε2 . Then yk0 ∈ I-st-LIMxr and so A = {n ∈ N : n1 |{k ≤ n : kxk − yk0 k ≥ r + ε2 }| ≥ δ} ∈ I. Since I is admissible so M = N \ A is nonempty. Choose n ∈ M . Then ε 1 |{k ≤ n : kxk − yk0 k ≥ r + } < δ n 2 ε 1 ⇒ |{k ≤ n : kxk − yk0 k < r + } ≥ 1 − δ. n 2 Put Bn = {k ≤ n : kxk − yk0 k < r + 2ε }. Then for k ∈ Bn , ε ε kxk − yk ≤ kxk − yk0 k + kyk0 − yk < r + + = r + ε. 2 2 Hence Bn ⊂ {k ≤ n : kxk − yk < r + ε}, which implies 1 − δ ≤ |Bnn | ≤ 1 1 n |{k ≤ n : kxk − yk < r + ε}|. Therefore n |{k ≤ n : kxk − yk ≥ r + ε}| < 1 − (1 − δ) = δ. Then 1 {n : |{k ≤ n : kxk − yk ≥ r + ε}| ≥ δ} ⊂ A ∈ I. n This shows that y ∈ I-st-LIMxr . Hence I-st-LIMxr is a closed set. Theorem 3.4. Let x = {xk }k∈N be sequence in X and r > 0 be a real number. Then the rough I-statistical limit set I-st-LIMxr of the sequence x is a convex set. Proof. Let y0 , y1 ∈ I-st-LIMxr and ε > 0 be given. Let A0 = {k ∈ N : kxk − y0 k ≥ r + ε} A1 = {k ∈ N : kxk − y1 k ≥ r + ε}. Then by theorem 3.1 for δ > 0 we have {n : n1 |{k ≤ n : k ∈ A0 ∪ A1 }| ≥ δ} ∈ I. Choose 0 < δ1 < 1 such that 0 < 1 − δ1 < δ. Let A = {n : n1 |{k ≤ n : k ∈ A0 ∪ A1 }| ≥ δ1 }. Then A ∈ I. Now for each n ∈ / A we have 1 |{k ≤ n : k ∈ A0 ∪ A1 }| < 1 − δ1 n 1 |{k ≤ n : k ∈ / A0 ∪ A1 }| ≥ {1 − (1 − δ1 )} = δ1 . ⇒ n


7

Therefore {k ∈ N : k ∈ / A0 ∪ A1 } is a nonempty set. Let us take k0 ∈ A0 c ∩ A1 c and 0 ≤ λ ≤ 1. Then kxk0 − (1 − λ)y0 − λy1 k =

k(1 − λ)xk0 + λxk0 − [(1 − λ)y0 + λy1 ]k

≤

(1 − λ)kxk0 − y0 k + λkxk0 − y1 k

0. Then a sequence x = {xk }k∈N in X is rough Istatistically convergent to ξ if and only if there exists a sequence y = {yk }k∈N in X such that I-st- lim y = ξ and k xk − yk k≤ r for all k ∈ N. Proof. Let y = {yk }k∈N be a sequence in X, which is I- statistically convergent to ξ and kxk − yk k ≤ r for all k ∈ N. Then for any ε > 0 and δ > 0 the set / A. Then A = {n ∈ N : n1 |{k ≤ n : kyk − ξk ≥ ε} ≥ δ} ∈ I. Let n ∈ 1 |{k ≤ n : kyk − ξk ≥ ε}| < δ n 1 ⇒ |{k ≤ n : kyk − ξk < ε} ≥ 1 − δ. n Let Bn = {k ≤ n : kyk − ξk < ε}, n ∈ N. Then for k ∈ Bn , we have kxk − ξk ≤ kxk − yk k + kyk − ξk < r + ε. Therefore, Bn ⊂ {k ≤ n : kxk − ξk < r + ε} |Bn | 1 ⇒ ≤ |{k ≤ n : kxk − ξk < r + ε}| n n 1 ⇒ |{k ≤ n : kxk − ξk < r + ε}| ≥ 1 − δ n 1 ⇒ |{k ≤ n : kxk − ξk ≥ r + ε}| < 1 − (1 − δ) = δ. n Therefore, {n ∈ N : n1 |{k ≤ n : kxk − ξk ≥ r + ε} ≥ δ} ⊂ A and since A ∈ I, we r -I -st have {n ∈ N : n1 |{k ≤ n : kxk − ξk ≥ r + ε} ∈ I. Hence xk → ξ.

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P. MALIK, M. MAITY AND AR. GHOSH r -I -st Conversely, suppose that xk −→ ξ. Then for ε > 0 and δ > 0,

A = {n ∈ N :

1 |{k ≤ n : kxk − ξk ≥ r + ε}| ≥ δ} ∈ I. n

Let n ∈ / A. Then 1 |{k ≤ n : kxk − ξk ≥ r + ε}| < δ. n 1 |{k ≤ n : kxk − ξk < r + ε}| ≥ 1 − δ. n Let Bn = {k ≤ n : kxk − ξk < r + ε}. Now we define a sequence y = {yk }k∈N as follows: ξ ,if kxk − ξk ≤ r, yk = k , otherwise. xk + r kxξ−x k −ξk ⇒

Then kyk − xk k = =

kξ − xk k ≤ r , if kxk − ξk ≤ r, r , otherwise.

Also, kyk − ξk =

0, k kxk − ξ + r kxξ−x k, k −ξk

=

0, kxk − ξk − r,

if kxk − ξk ≤ r otherwise.

if kxk − ξk ≤ r otherwise.

Let k ∈ Bn . Then kyk − ξk

=
0 dI ({k : kxk − λk < ε}) 6= 0 1 |{k n→∞ n

where dI (A) = I − lim

≤ n : k ∈ A}, if exists.

The set of I-statistical cluster point of x is denoted by ΛSx (I). Theorem 3.6. Let x = {xk }k∈N be a sequence in X and c ∈ ΛSx (I). Then kξ − ck ≤ r for all ξ ∈ I-st-LIMxr . Proof. If possible, let there exist ξ ∈ I-st-LIMxr such that kξ − ck > r. Let ε = kξ−ck−r . Then, 2 {k ∈ N : kxk − ξk ≥ r + ε} ⊃ {k ∈ N : kxk − ck < ε}.

(3)

Since c ∈ ΛSx (I), so dI ({k : kxk − ck < ε}) 6= 0. Hence by (3) we have dI ({k : kxk − ξk ≥ r + ε}) 6= 0, which contradicts that ξ ∈ I-st-LIMxr . Hence kξ − ck ≤ r. Definition 3.2. A sequence x = {xk }k∈N in X is said to be I- statistically bounded if there exists a positive number G such that for any δ > 0 the set A = {n ∈ N : n1 |{k ≤ n : kxk k ≥ G}| ≥ δ} ∈ I. Theorem 3.7. A sequence x = {xk }k∈N in X is I-statistically bounded if and only if there exists a non negative real number r > 0 such that I-st-LIMxr 6= ∅. Proof. Let x = {xk }k∈N be an I-statistically bounded sequence in X. Then there exists a positive real number G such that for δ > 0 we have {n : n1 |{k ≤ n : kxk k ≥ G}| ≥ δ} ∈ I. Let A = {k : kxk k ≥ G}. Then I- lim n1 |{k ≤ n : k ∈ n→∞

′

A}| = 0. Let r′ = sup{kxk k : k ∈ Ac }. Then the set I-st-LIMxr contains the origin. So we have I-st-LIMxr 6= ∅ for r = r′ . Conversely, let I-st-LIMxr 6= ∅ for some r > 0. Let ξ ∈ I-st-LIMxr . Choose ε = kξk. Then for each δ > 0, {n ∈ N : n1 |{k ≤ n : kxk − ξk ≥ r + ε}| ≥ δ} ∈ I. Now taking G = r + 2kξk, we have {n ∈ N : n1 |{k ≤ n : kxk k ≥ G}| ≥ δ} ∈ I. Therefore x is I-statistically bounded. Theorem 3.8. Let x = {xk }k∈N be a sequence in X and r > 0. (i) If c ∈ ΛSx (I), then I-st-LIMxr ⊂ Br (c). T Br (c) = {ξ ∈ X : ΛSx (I) ⊂ Br (ξ)} (ii) I-st-LIMxr = c∈ΛS x (I)

Proof. (i)Let c ∈ ΛSx (I). Then by Theorem 3.6, for all ξ ∈ I-st-LIMxr , kξ−ck ≤ r and hence the result follows.

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(ii) By (i) it is clear that I-st-LIMrx ⊂

T

Br (c). Now for all c ∈ ΛSx (I)

c∈ΛS x (I)

and y ∈

T

Br (c) we have ky − ck ≤ r. Then clearly

c∈ΛS x (I)

T

Br (c) ⊂ {ξ ∈ X :

c∈ΛS x (I)

ΛSx (I) ⊂ Br (ξ)}. Now, let y ∈ / I-st-LIMxr . Then there exists an ε > 0 such that dI (A) 6= 0, where A = {k ∈ N : kxk − yk ≥ r + ε} . This implies the existence of an I-statistical cluster point c of the sequence x with ky − ck ≥ r + ε. This gives ΛSx (I) * Br (y)and so y ∈ / {ξ ∈ X : ΛSx (I) ⊂ Br (ξ)}. Hence {ξ ∈ X : ΛSx (I) ⊂ r Br (ξ)} ⊂ I-st-LIMx . This completes the proof. Acknowledgment: The authors are grateful to Prof. Pratulananda Das, Department of Mathematics, Jadavpur University for his advice during the preparation of this paper. The third author is grateful to Government of India for his fellowship funding under UGC-JRF scheme during the preparation of this paper.

References [1] S. Aytar: Rough statistical covergence, Numer. Funct. Anal. And Optimiz. 29(3)(2008), 291-303. [2] P. Das, E. Savas S.K. Ghosal: On generalizations of certain summability methods using ideals, Appl. Math. Letters 24(2011), 1509-1514. [3] P. Das, E. Savas: On I -statistically precauchy sequences, Taiwanese J. of Math. 18(1)(Feb. 2014), 115-126. [4] P. Das, P. Malik:On the statistical and I-variation of double sequences, Tatra Mt. Math. Publ 40(2008), 91-112. [5] P. Das, P. Kostyrko, W. Wilczy´ nski, P. Malik : I and I ∗ -convergence of double sequences, Math. Slovaca 58(2008), 605-620. [6] J. A. Fridy: On statistical convergence, Analysis 5(1985), 301-313. [7] H. Fast: Surla convergence statistique, Colloq. Math. 2(1951), 241-244. ˇ at, W. Wilczy´ [8] P. Kostyrko, T. Sal´ nski: I -convergence, Real Anal. Exchange 26(2)(2000/2001), 669-685. [9] P. Malik, M. Maity: On rough statistical convergence of double sequences in normed linear spaces, Afr. Mat. 27(2016), 141-148 [10] P. Malik, M. Maity: On rough convergence of double sequences in normed linear spaces, Allahabad Mathematical Society 28(10(2013), 89-99. [11] P. Malik, M. Maity, A. Ghosh: A note on rough I-convergence of double sequences, arXiv preprint arXiv:1603.01363, 2016 [12] S.K. Pal, D. Chandra, S. Dutta: Rough ideal convergence, Hacettepe J. of Math. and Stat. 42(6)(2013), 633-640. [13] H.X. Phu: Rough convergence in normed linear spaces, Numer. Funct. Anal. And Optimiz. 22(2001), 201-224. [14] I. J. Schoenberg: The integrability of certain functions and related summability methods, Amer. Math. Monthly 66(1959) 361-375.


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ˇ at: On statistically convergent sequences of real numbers, Math. Slovaca 30(1980), [15] T. Sal´ 139-150. [16] E. Savas, P. Das: A generalized statistical convergence via ideals, App. Math. Letters 24(2011) 826-830. [17] U. Yamanc and M. G¯ urdal: I-Statistical convergence in 2-normed space, Arab J.Math.Sci. 20(1) (2014), 41-47.