axioms Article
New Definitons about AI -Statistical Convergence with Respect to a Sequence of Modulus Functions and Lacunary Sequences Ömer Ki¸si 1, *, Hafize Gümü¸s 2 and Ekrem Savas 3 1 2 3
*
Department of Mathematics, Faculty of Science, Bartin University, Bartin 74100, Turkey Department of Mathematics, Faculty of Eregli Education, Necmettin Erbakan University, Eregli, Konya 42060, Turkey;
[email protected] Department of Mathematics, Istanbul Ticaret University, Üsküdar, Istanbul 34840, Turkey;
[email protected] Correspondence:
[email protected]; Tel.: +90-378-501-1000
Received: 19 February 2018; Accepted: 10 April 2018; Published: 13 April 2018
Abstract: In this paper, using an infinite matrix of complex numbers, a modulus function and a lacunary sequence, we generalize the concept of I -statistical convergence, which is a recently introduced summability method. The names of our new methods are AI -lacunary statistical convergence and strongly AI -lacunary convergence with respect to a sequence of modulus functions. These spaces are denoted by SθA (I , F ) and NθA (I , F ) , respectively. We give some inclusion relations I
between S A (I , F ) , SθA (I , F ) and NθA (I , F ). We also investigate Cesáro summability for A and we I
I
obtain some basic results between A -Cesáro summability, strongly A -Cesáro summability and the spaces mentioned above. Keywords: lacunary sequence; statistical convergence; ideal convergence; modulus function; I -statistical convergence MSC: 40A35, 40A05
1. Introduction As is known, convergence is one of the most important notions in mathematics. Statistical convergence extends the notion. After giving the definition of statistical convergence, we can easily show that any convergent sequence is statistically convergent, but not conversely. Let E be a subset n 1 ∑ χ E ( j) n n → ∞ j =1
of N, and the set of all natural numbers d( E) := lim
is said to be a natural density of E
whenever the limit exists. Here, χ E is the characteristic function of E. In 1935, statistical convergence was given by Zygmund in the first edition of his monograph [1]. It was formally introduced by Fast [2], Fridy [3], Salat [4], Steinhaus [5] and later was reintroduced by Schoenberg [6]. It has become an active research area in recent years. This concept has applications in different fields of mathematics such as number theory [7], measure theory [8], trigonometric series [1], summability theory [9], etc. Following this very important definition, the concept of lacunary statistical convergence was defined by Fridy and Orhan [10]. In addition, Fridy and Orhan gave the relationships between the lacunary statistical convergence and the Cesàro summability. Freedman and Sember [9] established the connection between the strongly Cesàro summable sequences space |σ1 | and the strongly lacunary summable sequence space Nθ .
Axioms 2018, 7, 24; doi:10.3390/axioms7020024
www.mdpi.com/journal/axioms
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I -convergence has emerged as a generalized form of many types of convergences. This means that, if we choose different ideals, we will have different convergences. Koystro et al. [11] introduced this concept in a metric space. Also, Das et al. [12], Koystro et al. [13], Sava¸s and Das [14] studied ideal convergence. We will explain this situation with two examples later. Before defining I -convergence, the definitions of ideal and filter will be needed. An ideal is a family of sets I ⊆ 2N such that (i) ∅ ∈ I , (ii) A, B ∈ I implies A ∪ B ∈ I , (iii), and, for each A ∈ I , each B ⊆ A implies B ∈ I . An ideal is called non-trivial if N ∈ / I and a non-trivial ideal is called admissible if {n} ∈ I for each n ∈ N. A filter is a family of sets F ⊆ 2N such that (i) ∅ ∈ / F , (ii) A, B ∈ F implies A ∩ B ∈ F , (iii) For each A ∈ F , each A ⊆ B implies B ∈ F . If I is an ideal in N, then the collection F (I) = { A ⊂ N : N\ A ∈ I} forms a filter in N that is called the filter associated with I . The notion of a modulus function was introduced by Nakano [15]. We recall that a modulus f is a function from [0, ∞) to [0, ∞) such that (i) f ( x ) = 0 if and only if x = 0; (ii) f ( x + y) = f ( x ) + f (y) for x, y ≥ 0; (iii) f is increasing; and (iv) f is continuous from the right at 0. It follows that f must be continuous on [0, ∞) . Connor [16], Bilgin [17], Maddox [18], Kolk [19], Pehlivan and Fisher [20] and Ruckle [21] have used a modulus function to construct sequence spaces. Now, let S be the space of sequences of modulus functions F = ( f k ) such that limx→0+ supk f k ( x ) = 0. Throughout this paper, the set of all modulus functions determined by F is denoted by F = ( f k ) ∈ S for every k ∈ N. In this paper, we aim to unify these approaches and use ideals to introduce the notion of I A -lacunary statistically convergence with respect to a sequence of modulus functions. 2. Definitions and Notations First, we recall some of the basic concepts that will be used in this paper. Let A = ( aki ) be an infinite matrix of complex numbers. We write Ax = ( Ak ( x )), ∞
if Ak ( x ) = ∑ aki xk converges for each k. i =1
Definition 1. A number sequence x = ( xk ) is said to be statistically convergent to the number L if for every ε > 0, 1 lim |{k ≤ n : | xk − L| ≥ ε}| = 0. n→∞ n In this case, we write st − lim xk = L. As we said before, statistical convergence is a natural generalization of ordinary convergence i.e., if lim xk = L, then st − lim xk = L (Fast, [2] ). By a lacunary sequence, we mean an increasing integer sequence θ = {kr } such that k0 = 0 and hr = kr − kr−1 → ∞ as r → ∞. Throughout this paper, the intervals determined by θ will be denoted by Ir = (kr−1 , kr ]. Definition 2. A sequence x = ( xk ) is said to be lacunary statistically convergent to the number L if, for every ε > 0, 1 lim |{k ∈ Ir : | xk − L| ≥ ε}| = 0. r → ∞ hr In this case, we write Sθ − lim xk = L or xk → L(Sθ ) (Fridy and Orhan, [10] ).
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Definition 3. The sequence space Nθ is defined by ( Nθ =
1 ( xk ) : lim r → ∞ hr
∑ | xk − L| = 0
)
k ∈ Ir
(Fridy and Orhan, [10] ). Definition 4. Let I ⊂ 2N be a proper admissible ideal in N. The sequence ( xn ) of elements of R is said to be I -convergent to L ∈ R if, for each ε > 0, the set A (ε) = {n ∈ N : | xn − L| ≥ ε} ∈ I (Kostyrko et al. [11] ). Example 1. Define the set of all finite subsets of N by I f . Then, I f is a non-trivial admissible ideal and I f -convergence coincides with the usual convergence. Example 2. Define the set Id by Id = { A ⊂ N : d( A) = 0} . Then, Id is an admissible ideal and Id -convergence gives the statistical convergence. Following the line of Savas et al. [22], some authors obtained more general results about statistical convergence by using A matrix and they called this new method AI -statistical convergence (see, e.g., [17,23]). Definition 5. Let A = ( aki ) be an infinite matrix of complex numbers and ( f k ) be a sequence of modulus functions in S. A sequence x = ( xk ) is said to be AI -statistically convergent to L ∈ X with respect to a sequence of modulus functions, for each ε > 0, for every x ∈ X and δ > 0, n∈N:
1 |{k ≤ n : f k (| Ak ( x ) − L|) ≥ ε}| ≥ δ n
∈ I.
In this case, we write xk → L S A (I , F ) (Yamancıet al. [23] ). 3. Inclusions between S A (I , F ) , SθA (I , F ) and NθA (I , F ) Spaces We now consider our main results. We begin with the following definitions. Definition 6. Let A = ( aki ) be an infinite matrix of complex numbers, θ = {kr } be a lacunary sequence and F = ( f k ) be a sequence of modulus functions in S. A sequence x = ( xk ) is said to be AI -lacunary statistically convergent to L ∈ X with respect to a sequence of modulus functions, for each ε > 0, for each x ∈ X and δ > 0,
1 r ∈ N : | {k ∈ Ir : f k (| Ak ( x ) − L|) ≥ ε} | ≥ δ hr
∈ I.
Definition 7. Let A = ( aki ) be an infinite matrix of complex numbers, θ = {kr } be a lacunary sequence and F = ( f k ) be a sequence of modulus functions in S. A sequence x = ( xk ) is said to be strongly AI -lacunary convergent to L ∈ X with respect to a sequence of modulus functions, if, for each ε > 0, for each x ∈ X, (
1 r∈N: hr
∑
) f k (| Ak ( x ) − L|) ≥ ε
∈ I.
k ∈ Ir
We shall denote by SθA (I , F ), NθA (I , F ) the collections of all AI -lacunary statistically convergent and strongly AI -lacunary convergent sequences, respectively.
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Theorem 1. Let A = ( aki ) be an infinite matrix of complex numbers and ( f k ) be a sequence of modulus functions in S. SθA (I , F ) ∩ m ( X ) is a closed subset of m ( X ) if X is a Banach space where m( X ) is the space of all bounded sequences of X. Proof. Suppose that ( x n ) ⊂ SθA (I , F ) ∩ m ( X ) is a convergent sequence and it converges to x ∈ m ( X ). We need to show that x ∈ SθA (I , F ) ∩ m ( X ). Assume that x n → Ln SθA (I , F ) , ∀n ∈ N. Take a sequence {ε r }n∈N of strictly decreasing positive numbers converging to zero. We can find an
r ∈ N such that x − x j ∞ < ε4r for all j ≥ r. Choose 0 < δ < 51 . Now, 1 n ε r o n A= r∈N: < δ ∈ F (I) k ∈ Ir : f k (| Ak ( x ) − Ln |) ≥ hr 4 and
B=
1 n ε r o n +1 r∈N: − L n +1 | ≥ < δ ∈ F (I) . k ∈ Ir : f k | Ak x hr 4
Since A ∩ B ∈ F (I) and ∅ ∈ / F (I), we can choose r ∈ A ∩ B. Then, 1 n εr ε r o ∨ f k | A k x n +1 − L n +1 | ≥ ≤ 2δ < 1. k ∈ Ir : f k (| Ak ( x n ) − Ln |) ≥ hr 4 4 Since hr → ∞ and A ∩ B ∈ F (I) is infinite, we can actually choose the above r so that n εr hr > 5. Hence, there must exist a k ∈ Ir for which we have simultaneously, xk − Ln < 4 and n +1 xk − Ln+1 < ε4r . Then, it follows that | Ln − Ln+1 | ≤ Ln − xkn + xkn − xkn+1 + xkn+1 − Ln+1
≤ xkn − Ln + xkn+1 − Ln+1 + k x − x n k∞ + x − x n+1 ∞
≤
εr 4
+
εr 4
+
εr 4
+
εr 4
= εr.
This implies that { Ln }n∈N is a Cauchy sequence in X. Since X is a Banach space, we can write Ln → L ∈ X as n → ∞. We shall prove that xk → L SθA (I , F ) . Choose ε > 0 and r ∈ N such that ε r < 4ε , k x − xn k∞ < 4ε . Now, since 1 hr
|{k ∈ Ir : f k (| Ak ( x ) − L|) ≥ ε}|
≤
1 hr
|{k ∈ Ir : f k (| Ak ( x − xn ) |) + f k (| Ak ( x n ) − Ln |) + f k (| Ln − L|) ≥ ε}|
≤
1 hr
k ∈ Ir : f k (| Ak ( x n ) − Ln |) ≥ ε . 2
It follows that n
r∈N:
1 hr
n ⊂ r∈N:
|{k ∈ Ir : f k (| Ak ( x ) − L|) ≥ ε}| ≥ δ 1 hr
o
o k ∈ Ir : f k (| Ak ( x ) − L|) ≥ ε ≥ δ 2
for given δ > 0. This shows that xk → L SθA (I , F ) and this completes the proof of the theorem. Theorem 2. Let A = ( aki ) be an infinite matrix of complex numbers, θ = {kr } be a lacunary sequence and ( f k ) be a sequence of modulus functions in S. Then, we have
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(i) If xk → L NθA (I , F ) , then xk → L SθA (I , F ) and NθA (I , F ) ⊂ SθA (I , F ) is proper for every ideal I ; (ii) If x ∈ m ( X ), the space of all bounded sequences of X and xk → L SθA (I , F ) , then xk → L NθA (I , F ) ; (iii) SθA (I , F ) ∩ m ( X ) = NθA (I , F ) ∩ m ( X ) . Proof. (i) Let ε > 0 and xk → L NθA (I , F ) . Then, we can write ∑ f k (| Ak ( x ) − L|)
k ∈ Ir
≥
∑
k∈ Ir f k (| Ak ( x )− L|)≥ε
f k (| Ak ( x ) − L|)
≥ ε |{k ∈ Ir : f k (| Ak ( x ) − L|) ≥ ε}| . Thus, for given δ > 0, 1 1 |{k ∈ Ir : f k (| Ak ( x ) − L|) ≥ ε}| ≥ δ =⇒ hr hr
∑
f k (| Ak ( x ) − L|) ≥ εδ,
k ∈ Ir
i.e.,
1 r∈N: |{k ∈ Ir : f k (| Ak ( x ) − L|) ≥ ε}| ≥ δ hr
(
⊆
1 r∈N: hr
∑
) f k (| Ak ( x ) − L|) ≥ εδ
.
k ∈ Ir
Since xk → L NθA (I , F ) , the set on the right-hand side belongs to I and so it follows that xk → L SθA (I , F ) . To show that SθA (I , F ) NθA (I , F ) , take a fixed K ∈ I . Define x = ( xk ) by √ / K, ku, for kr−1 < k ≤ kr−1 + √hr , r = 1, 2, 3..., r ∈ ( xk ) = ku, for kr−1 < k ≤ kr−1 + hr , r = 1, 2, 3..., r ∈ K, θ, otherwise, where u ∈ X is a fixed element with kuk = 1 and θ is the null element of X. Then, x ∈ / m ( X ) and for every 0 < ε < 1 since √ hr 1 → 0. |{k ∈ Ir : f k (| Ak ( x ) − 0|) ≥ ε}| = √ hr hr As r → ∞ and r ∈ / K, for every δ > 0, r∈N:
1 | {k ∈ Ir : f k (| Ak ( x ) − 0|) ≥ ε} | ≥ δ hr
⊂ M ∪ {1, 2, ..., m}
for some m ∈ N. Since I is admissible, it follows that xk → θ SθA (I , F ) . Obviously, 1 hr
∑
f k (| Ak ( x ) − θ |) → ∞,
k ∈ Ir
i.e., xk 9 θ NθA (I , F ) . Note that, if K ∈ I is finite, then xk 9 θ SθA . This example shows that AI -lacunary statistical convergence is more general than lacunary statistical convergence. (ii) Suppose that x ∈ l∞ and xk → L SθA (I , F ) . Then, we can assume that f k (| Ak ( x ) − L|) ≤ M for each x ∈ X and all k.
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Given ε > 0, we get 1 ∑ f (| Ak ( x ) − L|) hr k∈ Ir k
=
1 hr
+ ≤
∑
k ∈ Ir f k (| Ak x − L|)≥ε
1 hr
∑
f k (| Ak ( x ) − L|)
k∈ Ir f k (| Ak x − L|) 1, then xk → L S A (I , F ) ⇒ xk → L SθA (I , F ) . Proof. Suppose first that lim infr qr > 1, then there exists δ > 0 such that qr ≥ 1 + δ for sufficiently large r, which implies that hr δ ≥ . kr 1+δ If xk → L SθA (I , F ) , then for every ε > 0, for each x ∈ X and for sufficiently large r, we have 1 |{k ≤ kr : f k (| Ak ( x ) − L|) ≥ ε}| kr
≥
1 |{k ∈ Ir : f k (| Ak ( x ) − L|) ≥ ε}| kr
≥
δ 1 |{k ∈ Ir : f k (| Ak ( x ) − L|) ≥ ε}| . 1 + δ hr
Then, for any δ > 0, we get n
r∈N:
1 hr
n ⊆ r∈N:
|{k ∈ Ir : f k (| Ak ( x ) − L|) ≥ ε}| ≥ δ 1 kr
o
|{k ≤ kr : f k (| Ak ( x ) − L|) ≥ ε}| ≥
δα ( α +1)
o
∈ I.
This completes the proof. For the next result, we assume that the lacunary sequence θ satisfies the condition that, for any set C ∈ F (I), ∪ {n : kr−1 < n ≤ kr , r ∈ C } ∈ F (I).
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Theorem 4. Let A = ( aki ) be an infinite matrix of complex numbers and ( f k ) be a sequence of modulus functions in S. If θ = {kr } is a lacunary sequence with lim supr qr < ∞, then xk → L SθA (I , F ) implies xk → L S A (I , F ) . Proof. If lim supr qr < ∞, then, without any loss of generality, we can assume that there exists a 0 < M < ∞ such that qr < M for all r ≥ 1. Suppose that xk → L SθA (I , F ) , and for ε, δ, δ1 > 0 define the sets 1 C= r∈N: |{k ∈ Ir : f k (| Ak ( x ) − L|) ≥ ε}| < δ hr and
T=
n∈N:
1 |{k ≤ n : f k (| Ak ( x ) − L|) ≥ ε}| < δ1 . n
It is obvious from our assumption that C ∈ F (I), the filter associated with the ideal I . Further observe that 1 Kj = k ∈ Ij : f k (| Ak ( x ) − L|) ≥ ε < δ hj for all j ∈ C. Let n ∈ N be such that kr−1 < n ≤ kr for some r ∈ C. Now, 1 |{k ≤ n : f k | Ak ( x ) − L| ≥ ε}| n
≤
1 |{k ≤ kr : f k (| Ak ( x ) − L|) ≥ ε}| k r −1
=
1 {|{k ∈ I1 : f k (| Ak ( x ) − L|) ≥ ε}|} k r −1
+
1 {|{k ∈ I2 : f k (| Ak x − L|) ≥ ε}|} k r −1
+... +
=
k1 1 |{k ∈ I1 : f k | Ak ( x ) − L| ≥ ε}| k r −1 h 1
+
k2 − k1 1 |{k ∈ I2 : f k (| Ak ( x ) − L|) ≥ ε}| k r −1 h 2
+... +
=
≤
1 {|{k ∈ Ir : f k | Ak ( x ) − L| ≥ ε}|} k r −1
k r − k r −1 1 |{k ∈ Ir : f k (| Ak ( x ) − L|) ≥ ε}| k r −1 h r
k1 k2 − k1 k r − k r −1 K + K2 + ... + Kr k r −1 1 k r −1 k r −1 n
sup j∈C K j
o k r < Mδ. k r −1
δ and in view of the fact that ∪ {n : kr−1 < n ≤ kr , r ∈ C } ⊂ T where C ∈ F (I) , M it follows from our assumption on θ that the set T also belongs to F (I) and this completes the proof of the theorem. Choosing δ1 =
Combining Theorems 3 and 4, we get the following theorem.
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Theorem 5. Let A = ( aki ) be an infinite matrix of complex numbers and ( f k ) be a sequence of modulus functions in S. If θ = {kr } is a lacunary sequence with 1 < lim infr qr ≤ lim supr qr < ∞, then xk → L SθA (I , F ) = xk → L SθA (I , F ) . 4. Cesàro Summability for AI Definition 8. Let A = ( aki ) be an infinite matrix of complex numbers and ( f k ) be a sequence of modulus functions in S. A sequence x = ( xk ) is said to be AI -Cesàro summable to L if, for each ε > 0 and for each x ∈ X, ) ( 1 n n ∈ N : ∑ f k ( Ak ( x ) − L) ≥ ε ∈ I . n k =1 In this case, we write xk → L (σ1 )θA (I , F ) . Definition 9. Let A = ( aki ) be an infinite matrix of complex numbers and ( f k ) be a sequence of modulus functions in S. A sequence x = ( xk ) is said to be strongly AI -Cesàro summable to L if, for each ε > 0 and for each x ∈ X, ( ) 1 n n ∈ N : ∑ f k (| Ak ( x ) − L|) ≥ ε ∈ I . n k =1 In this case, we write xk → L |σ1 |θA (I , F ) . Theorem 6. Let θ be a lacunary sequence. If lim infr qr > 1, then xk → L |σ1 |θA (I , F ) ⇒ xk → L NθA (I , F ) . Proof. If lim infr qr > 1, then there exists δ > 0 such that qr ≥ 1 + δ for all r ≥ 1. Since hr = kr − kr−1 , kr k 1+δ 1 we have and r−1 ≤ . Let ε > 0 and define the set ≤ hr δ hr δ ) ( 1 kr S = kr ∈ N : f k (| Ak ( x ) − L|) < ε . kr k∑ =1 We can easily say that S ∈ F (I), which is a filter of the ideal I , 1 hr
∑ f k (| Ak ( x ) − L|)
1 hr
=
k ∈ Ir
≤
for each kr ∈ S. Choose η = (
and it completes the proof.
1+δ δ
k =1
kr 1 hr k r
=
kr
∑ f k (| Ak ( x ) − L|) −
1 hr
kr
∑ f k (| Ak ( x ) − L|) −
k =1
1+δ δ
k r −1
∑ f k (| Ak ( x ) − L|)
k =1
k r −1 1 h r k r −1
k r −1
∑ f k (| Ak ( x ) − L|)
k =1
1 ε − ε0 δ
1 ε − ε0 . Therefore, δ
1 r∈N: hr
∑
k ∈ Ir
) f k (| Ak ( x ) − L|) < η
∈ F (I),
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Theorem 7. Let A = ( aki ) be an infinite matrix of complex numbers and ( f k ) be a sequence of modulus A functions in S. If ( xk ) ∈ m ( X ) and xk → L Sθ (I , F ) , then xk → L (σ1 )θA (I , F ) . Proof. Suppose that ( xk ) ∈ m ( X ) and xk → L SθA (I , F ) . Then, we can assume that f k (| Ak x − L|) ≤ M for all k ∈ N. In addition, for each ε > 0, we can write n 1 n ∑ f k ( Ak ( x ) − L) ≤ 1 ∑ f k (| Ak ( x ) − L|) n n k =1 k =1 n
1 n
≤
∑
k =1
f k (| Ak ( x ) − L|)
f k (| Ak ( x )− L|)≥ 2ε
+
1 n
n
∑
k =1
f k (| Ak ( x ) − L|)
f k (| Ak ( x )− L|)< 2ε
1 M |{k ≤ n : f k (| Ak ( x ) − L|) ≥ ε}| + ε. n
≤
Consequently, if δ > ε > 0, δ and ε are independent, and, putting δ1 = δ − ε > 0, we have (
) 1 n n∈N: ∑ f ( A ( x ) − L) ≥ δ n k,l =1 k k
n ⊆ n ∈ N : n1 |{k ≤ n : f k (| Ak ( x ) − L|) ≥ ε}| ≥
δ1 M
o
∈ I.
This shows that xk → L (σ1 )θA (I , F ) . Theorem 8. Let θ be a lacunary sequence. If lim supr qr < ∞, then xk → L NθA (I , F ) ⇒ xk → L |σ1 |θA (I , F ) . Proof. If lim supr qr < ∞, then there exists M > 0 such that qr < M for all r ≥ 1. Let xk → L NθA (I , F ) and define the sets T and R such that ( T= and
( R=
1 r∈N: hr
1 n∈N: n
Let Aj =
1 hj
∑
) f k (| Ak ( x ) − L|) < ε 1
k ∈ Ir
)
n
∑
f k (| Ak ( x ) − L|) < ε 2
k =1
∑
k∈ Ij
f k (| Ak ( x ) − L|) < ε 1
.
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for all j ∈ T. It is obvious that T ∈ F (I). Choose n as being any integer with kr−1 < n < kr , where r ∈ T, 1 n
n
∑ f k (| Ak ( x ) − L|)
≤
k =1
=
kr
1
k r −1
∑ f k (| Ak ( x ) − L|)
k =1
1 k r −1
∑ f k (| Ak ( x ) − L|) + ∑ f k (| Ak ( x ) − L|)
k ∈ I1
k ∈ I2
!
+... + ∑ f k (| Ak ( x ) − L|) k ∈ Ir
!
!
=
k1 k r −1
1 h1
+... +
k r − k r −1 k r −1
=
k1 k r −1
≤
∑ f k (| Ak ( x ) − L|)
+
k ∈ I1
A1 +
1 h2
∑ f k (| Ak ( x ) − L|)
k ∈ I2
! 1 hr
k 2− k 1 k r −1
sup j∈T A j
k 2− k 1 k r −1
∑ f k (| Ak ( x ) − L|)
k ∈ Ir
A2 + ... +
k r − k r −1 k r −1
Ar
k1 k r −1
< ε 1 M. ε1 Choose ε 2 = M and in view of the fact that ∪ {n : kr−1 < n < kr , r ∈ T } ⊂ R, where T ∈ F (I), ıt follows from our assumption on θ that the set R also belongs to F (I) and this completes the proof of the theorem.
Theorem 9. If xk → L |σ1 |θA (I , F ) , then xk → L S A (I , F ) . Proof. Let xk → L |σ1 |θA (I , F ) and ε > 0 is given. Then, n
∑ f k (| Ak ( x ) − L|)
n
∑
≥
k =1
k =1
f k (| Ak ( x ) − L|)
f k (| Ak x − L|)≥ε
≥ ε |{k ≤ n : f k (| Ak ( x ) − L|) ≥ ε}| and so
1 εn
n
∑
f k (| Ak ( x ) − L|) ≥
k =1
1 |{k ≤ n : f k (| Ak ( x ) − L|) ≥ ε}| . n
Thus, for a given δ > 0,
1 n ∈ N : |{k ≤ n : f k (| Ak ( x ) − L|) ≥ ε}| ≥ δ n
⊆ Therefore, xk → L S A (I , F ) .
n∈N:
1 n ∑ f (| Ak ( x ) − L|) ≥ εδ n k =1 k
∈ I.
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Theorem 10. Let ( xk ) ∈ m ( X ). If xk → L S A (I , F ) . Then, xk → L |σ1 |θA (I , F ) . Proof. Suppose that ( xk ) is bounded and xk → L S A (I , F ) . Then, there is an M such that f k (| Ak ( x ) − L|) ≤ M for all k. Given ε > 0, we have 1 n ∑ f (| Ak ( x ) − L|) n k =1 k
=
n
1 n
∑
k =1
f k (| Ak ( x ) − L|)
f k (| Ak ( x )− L|)≥ε
+
n
1 n
∑
k =1
f k (| Ak ( x ) − L|)
f k (| Ak ( x )− L|) 0,
1 n n∈N: ∑ f (| Ak ( x ) − L|) ≥ δ n k =1 k
⊆
n∈N:
δ 1 |{k ≤ n : f k (| Ak ( x ) − L|) ≥ ε}| ≥ n M
∈ I.
Therefore, xk → L |σ1 |θA (I , F ) . 5. Conclusions
I -statistical convergence gained a different perspective after identification of the AI -statistical convergence with an infinite matrix of complex numbers. Some authors have studied this new method with different sequences. Our results in this paper were developed with lacunary sequences. By also using a modulus function, we obtain more interesting and general results. These definitions can be adapted to many different concepts such as random variables in order to have different results. Acknowledgments: We would like to thanks the referees for their valuable comments. Author Contributions: Both authors completed the paper together. Both authors read and approved the final manuscript. Conflicts of Interest: The authors declare no conflict of interest..
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