λ-Statistical convergence of order α

Acta Mathematica Scientia 2011,31B(3):953–959 http://actams.wipm.ac.cn

λ-STATISTICAL CONVERGENCE OF ORDER α∗ R. C ¸ olak

C ¸ . A. Bekta¸s

Department of Mathematics, Firat University, 23119, Elazı˘ g T¨ urkiye E-mail: [email protected]; [email protected]

Abstract In this paper, we introduce the concept of λ-statistical convergence of order α. Also some relations between the λ-statistical convergence of order α and strong (V, λ)summability of order α are given. Key words sequences; statistical convergence; Ces` aro summability 2000 MR Subject Classification

1

40A35; 40A05; 40C05; 46A45

Introduction

The idea of statistical convergence was given by Zygmund [22] in the first edition of his monograph published in Warsaw in 1935. The concept of statistical convergence was introduced by Steinhaus [19] and Fast [5] and later reintroduced by Schoenberg [18] independently. Over the years and under different names statistical convergence was discussed in the theory of Fourier analysis, ergodic theory, number theory, measure theory, trigonometric series, turnpike theory and Banach spaces. Later on it was further investigated from the sequence space point of view and linked with summability theory by Fridy [6], Connor [3], Sava¸s [17], Mursaleen [12], Fridy and Orhan [7], M´ oricz [11], Rath and Tripathy [14], Salat [16], Bhardwaj [1] and many others. In recent years, generalizations of statistical convergence appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions on locally compact spaces. Statistical convergence and its generalizations were also connected ˇ with subsets of the Stone-Cech compactification of the natural numbers. Moreover, statistical convergence is closely related to the concept of convergence in probability. A sequence x = (xk ) is said to be statistical convergent to the number L if, for every ε > 0, lim

n→∞

1 |{k ≤ n : |xk − L| ≥ ε}| = 0, n

where the vertical bars indicate the number of elements in the enclosed set. In this case, we write S −lim x = L or xk → L(S) and S denotes the set of all statistically convergent sequences. The statistical convergence with degree 0 < β < 1 was given by Gadjiev and Orhan [8] for a number sequence, and then it was generalized for the method of A-statistical convergence ∗ Received

June 15, 2009; revised November 4, 2009. This research was supported by FUBAP under the Project no. 1683.

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by Duman et.al., [4]. After then the statistical convergence of order α and strong p-Ces`aro summability of order α were studied by C ¸ olak [2] for number sequences. Let 0 < α ≤ 1 be given. The sequence (xk ) is said to be statistically convergent of order α if there is a complex number L such that lim

n→∞

1 |{k ≤ n : |xk − L| ≥ ε}| = 0 nα

for every ε > 0, in which case we say that x is statistically convergent to L of order α (see [2]). This definition is same with Definition 4 in [8] if we take α = 1 − β, 0 < β < 1. In this case we write S α − lim xk = L. The set of all statistically convergent sequences of order α will be denoted by S α . We write Soα to denote the set of all statistically null sequences of order α. It is clear that Soα ⊂ S α for each 0 < α ≤ 1. Throughout the paper, w and c will denote the spaces of all and convergent sequences of complex numbers, respectively, and C will denote the set of complex number. Let λ = (λn ) be a non-decreasing sequence of positive numbers tending to ∞ such that λn+1 ≤ λn + 1, λ1 = 1. The generalized de la Vall´ee-Pousin mean is defined by tn (x) =

1  xk , λn k∈In

where In = [n − λn + 1, n]. A sequence x = (xk ) is said to be (V, λ)-summable to a number L (see [9]) if tn (x) → L as n → ∞. If λn = n for each n ∈ N, then (V, λ)-summability reduces to (C, 1)-summability. We write  [C, 1] =

 n 1 x = (xk ) ∈ w : ∃L ∈ C, lim |xk − L| = 0 , n→∞ n k=1

 [V, λ] =

1  |xk − L| = 0 x = (xk ) ∈ w : ∃L ∈ C, lim n→∞ λn



k∈In

for the sets of sequences x = (xk ) which are strongly Ces`aro summable and strongly (V, λ)summable to L, i.e., xk → L [C, 1] and xk → L [V, λ], respectively. A sequence x = (xk ) is said to be λ-statistically convergent or Sλ -convergent to L if for every ε > 0 1 lim |{k ∈ In : |xk − L| ≥ ε}| = 0. n→∞ λn In this case we write Sλ −lim x = L or xk → L(Sλ ), and Sλ = {x = (xk ) ∈ w : ∃L ∈ C, Sλ − lim x = L} (see [12]).

R. C ¸ olak & C ¸ . A. Bekta¸s: λ-STATISTICAL CONVERGENCE OF ORDER α

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Main Results

In this section we give the main results of the paper. In Theorem 2.4, we give the inclusion relations between the sets of λ-statistical convergent sequences of order α for different α s, and so that the inclusion relations between the set of λ-statistical convergent sequences of order α and the set of λ-statistical convergent sequences. In Theorem 2.9, we give the relationship between the strong p-Ces`aro summability of order α and the strong p-Ces`aro summability of order β. In Theorem 2.12, we give the relationship between the strong p-Ces`aro summability of order α and the λ-statistical convergence of order β. Definition 2.1 Let the sequence λ = (λn ) of real numbers be defined as above and 0 < α ≤ 1 be given. The sequence x = (xk ) ∈ w is said to be λ-statistically convergent of order α if there is a complex number L such that lim

n→∞

1 |{k ∈ In : |xk − L| ≥ ε}| = 0, λα n α

α α where In = [n − λn + 1, n] and λα n denote the αth pover (λn ) of λn , that is, λ = (λn ) = α α α (λα 1 , λ2 , · · · , λn , · · ·) . In this case we write Sλ − lim xk = L. The set of all λ-statistically convergent sequences of order α will be denoted by Sλα . We α write Sλ,o to denote the set of all λ-statistically null sequences of order α. α ⊂ Sλα for each 0 < α ≤ 1. The λ-statistical convergence of order It is clear that Sλ,o α is same with the λ-statistical convergence, that is Sλα = Sλ for α = 1. The λ-statistical convergence of order α is well defined for 0 < α ≤ 1. But it is not well defined for α > 1 in general. This follows from the following example. Example 1 Let x = (xk ) be defined as follows: ⎧ ⎨ 1, k = 2n n = 1, 2, 3, · · · . xk = ⎩ 0, k = 2n

Then both lim

1 [λn ] + 1 |{k ∈ In : |xk − 1| ≥ ε}| ≤ lim =0 n→∞ λα 2λα n n

lim

1 [λn ] + 1 |{k ∈ In : |xk − 0| ≥ ε}| ≤ lim =0 α n→∞ λn 2λα n

n→∞

and n→∞

for α > 1, such that x = (xk ) λ-statistically converges, of order α, both to 1 and 0, i.e., Sλα − lim xk = 1 and Sλα − lim xk = 0. But this is impossible. Remark Let λ = (λn ) be defined as above, 0 < α ≤ 1 and In (α) = [n − λα n + 1, n] . Let us define 1 x = (xk ) ∈ Sλ(α) ⇐⇒ lim α |{k ∈ In (α) : |xk − L| ≥ ε}| = 0. n→∞ λn α Since λα n ≤ λn , we have Sλ ⊂ Sλ(α) for 0 < α ≤ 1. Now Sλ(α) coincides with Sμ with μ = (μn ) = (λα n ) , given by Mursaleen [12]. Theorem 2.2 Let 0 < α ≤ 1 and x = (xk ), y = (yk ) be sequences of complex numbers. (i) If Sλα − lim xk = xo and c ∈ C, then Sλα − lim(cxk ) = cxo . (ii) If Sλα − lim xk = xo and Sλα − lim yk = y0 , then Sλα − lim(xk + yk ) = xo + yo .

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(i) It is clear in case c = 0. Suppose that c = 0, then the proof of (i) follows from   1 1  ε  |{k ∈ In : |cxk − cxo | ≥ ε}| = α {k ∈ In : |xk − xo | ≥ } λα λn |c| n

and that of (ii) follows from 1 λα n 1 ≤ α λn

|{k ∈ In : |xk + yk − (xo + yo )| ≥ ε}|  1  ε  ε    + α  k ∈ In : |yk − yo | ≥ .  k ∈ In : |xk − xo | ≥ 2 λn 2

Note that Sλα is different from Sλ defined in [12] in general. If we take λn = nα for 0 < α < 1, then S α ⊆ Sλ . If λn = nα with α = 1, that is, λn = n and α = 1, then S α = Sλ = S (see [12]), that is, the statistical convergence of order α, statistical convergence and λ-statistical convergence coincide when λn = nα with α = 1. It is easy to see that every convergent sequence is statistically convergent of order α, that is, c ⊂ S α for each 0 < α ≤ 1. But it follows from the following example that the converse does not hold. Example 2 The sequence x = (xk ) defined by ⎧ ⎨ 1, k = n3 , xk = ⎩ 0, k = n3 is statistically convergent of order α with S α − lim xk = 0 for α > 13 , but it is not convergent. Definition 2.3 Let λ = (λn ) be a non-decreasing sequence of positive numbers tending to ∞ such that λn+1 ≤ λn + 1, λ1 = 1. Let α be any real number such that 0 < α ≤ 1 and let p be a positive real number. A sequence x is said to be strongly (V, λ)-summable of order α if there is a complex number L such that 1  p lim α |xk − L| = 0, n→∞ λn k∈In

where In = [n − λn + 1, n] . In such case we say that x is strong (V, λ)-summable of order α, to L. The strong (V, λ)-summability of order α is reduced to the strong (V, λ)-summability for  α = 1. The set of all strong (V, λ)-summable sequences of order α will be denoted by ωpα . We  α write ωop in case L = 0. Theorem 2.4 Let 0 < α ≤ β ≤ 1. Then Sλα ⊆ Sλβ and the inclusion is strict for some α and β such that α < β. Proof If 0 < α ≤ β ≤ 1, then 1 λβn

|{k ∈ In : |xk − L| ≥ ε}| ≤

1 |{k ∈ In : |xk − L| ≥ ε}| λα n

for every ε > 0, which gives that Sλα ⊆ Sλβ . It follows from the following example that the inclusion is strict. Example 3 Consider the sequence x = (xk ) defined by ⎧ ⎨ k, n − √λ + 1 ≤ k ≤ n, n xk = ⎩ 0, otherwise.

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Then Sλβ − lim xk = 0, i.e., x ∈ Sλβ for 12 < β ≤ 1 but x ∈ / Sλα for 0 < α ≤ 12 . If we take β = 1 in Theorem 2.4, then we obtain the following result. Corollary 2.5 If a sequence is λ-statistically convergent to L of order α, then it is λ-statistically convergent to L, that is, Sλα ⊆ Sλ for each α ∈ (0, 1], and the inclusion is strict. The following result is a consequence of Theorem 2.4. Corollary 2.6 (i) Sλα = Sλβ ⇐⇒ α = β. (ii) Sλα = Sλ ⇐⇒ α = 1. Theorem 2.7 Sλα ⊆ S for all λ and each α ∈ (0, 1]. Proof It is easily seen that Sλ ⊆ S for all λ, since λn /n is bounded by 1 (Mursaleen [12]). We know that Sλα ⊆ Sλ from Corollary 2.5. Then we have Sλα ⊆ S. Theorem 2.8 S ⊆ Sλα if and only if lim inf n→∞

Proof

λα n > 0. n

(1)

For a given ε > 0, we have {k ≤ n : |xk − L| ≥ ε} ⊃ {k ∈ In : |xk − L| ≥ ε} .

Therefore, 1 1 |{k ≤ n : |xk − L| ≥ ε}| ≥ |{k ∈ In : |xk − L| ≥ ε}| n n λα 1 = n · α |{k ∈ In : |xk − L| ≥ ε}| . n λn Taking limit as n → ∞ and using (1), we get xk → L(S) =⇒ xk → L(Sλα ). Conversely, suppose that lim inf that

λα n(j) n(j)

n→∞

λα n n

∞

= 0. We can choose a subsequence (n(j))j=1 such

< 1j . Define a sequence x = (xi ) by xi =

⎧ ⎨ 1,

i ∈ In(j) , j = 1, 2, · · · ,

⎩ 0,

otherwise.

Then x ∈ S, but on the other hand, x ∈ / Sλ . From Corollary 2.5, since Sλα ⊆ Sλ , we have x∈ / Sλα . Hence (1) is necessary.   Theorem 2.9 Let 0 < α ≤ β ≤ 1 and p be a positive real number. Then ωpα ⊆ ωpβ and the inclusion is strict for some α and β such that α < β.  Proof Let x = (xk ) ∈ ωpα . Then given α and β such that 0 < α ≤ β ≤ 1 and a positive real number p, we may write 1  λβn

k∈In

1  p |xk − L| λα n k∈In

 which gives ωp ⊆ ωpβ . To show that the inclusion is strict, consider the following example. 

α

p

|xk − L| ≤

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Example 4

Consider the sequence x = (xk ) defined by ⎧ ⎨ 1, n − √λ + 1 ≤ k ≤ n, n xk = ⎩ 0, otherwise.

It is easy to see that 1  λβn

Since since

1

β−1/2

λn

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√

p

|xk − 0| ≤

k∈In

λn

λβn

=

1 β−1/2 λn

.

  → 0 as n → ∞, then ωpβ − lim xk = 0, i.e., x ∈ ωpβ for

1 2

< β ≤ 1; but

√ λn − 1 1  p ≤ α |xk − 0| α λn λn k∈In √  α λn −1 and λα → ∞ as n → ∞, then x ∈ / ωp for 0 < α < 12 . This completes the proof. n The following result is a consequence of Theorem 2.9. Corollary 2.10 Let 0 < α ≤ β ≤ 1 and p be a positive real number. Then   (i) ωpα = ωpβ if and only if α = β.  (ii) ωpα ⊆ [ωp ] for each α ∈ (0, 1] and 0 < p < ∞.   Theorem 2.11 Let 0 < α ≤ 1 and 0 < p < q < ∞. Then ωqα ⊂ ωpα . Its proof is seen from H¨older inequality. Theorem 2.12 Let α and β be fixed real numbers such that 0 < α ≤ β ≤ 1 and 0 < p < ∞. If a sequence is strongly (V, λ)-summable to L of order α, then it is λ-statistically  convergent to L of order β, i.e., ωpα ⊂ Sλβ . Proof For any sequence x = (xk ) and ε > 0, we have    p p p |xk − L| = |xk − L| + |xk − L| k∈In

k∈In

k∈In

|xk −L|≥ε

≥



|xk −L|<ε p

|xk − L| ≥ |{k ∈ In : |xk − L| ≥ ε}| · εp

k∈In

|xk −L|≥ε

and so

1  1 p |xk − L| ≥ α |{k ∈ In : |xk − L| ≥ ε}| · εp λα λ n n k∈In

≥

1 λβn

|{k ∈ In : |xk − L| ≥ ε}| · εp .

From this it follows that if x = (xk ) is strong (V, λ)-summable to L of order α, then it is λ-statistically convergent to L of order β. If we take α = β in Theorem 2.12, we obtain the following result. Corollary 2.13 Let α be a fixed real number such that 0 < α ≤ 1 and 0 < p < ∞. If a sequence is strong (V, λ)-summable to L of order α, then it is λ-statistically convergent to L of  order α, i.e., ωpα ⊂ Sλα .  Corollary 2.14 Let 0 < α ≤ 1 and p be a positive real number. Then ωpα ⊂ Sλ and the inclusion is strict if 0 < α < 1.  Proof From Corollary 2.5 and Corollary 2.13, we have ωpα ⊂ Sλ .

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