Available at: http://www.pmf.ni.ac.rs/filomat
Some Cesaro-Type Summability Spaces of Order
α and Lacunary
Statistical Convergence of Order
α
Mikail Eta,b, Hacer S¸eng ¨ula
aDepartment of Mathematics ; Fırat University 23119 ; Elazıg ; TURKEY bDepartment of Mathematics ; Siirt University 56100 ; Siirt ; TURKEY
Abstract. In the paper [32], we have defined the concepts of lacunary statistical convergence of order α and strong Nθ p −summability of orderα for sequences of complex (or real) numbers. In this paper we continue to examine others relations between lacunary statistical convergence of orderα and strong Nθ p −summability of orderα.
1. Introduction
The idea of statistical convergence was given by Zygmund [34] in the first edition of his monograph puplished in Warsaw in 1935. The consept of statistical convergence was introduced by Steinhaus [33] and Fast [10] and later reintroduced by Schoenberg [31] independently. Over the years and under different names statistical convergence was discussed in the theory of Fourier analysis, ergodik 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 Bhardwaj and Bala [2], C¸ olak ([3],[4]), Connor [5], Et et al. [9], Fridy [12], Fridy and Orhan ([13], [14]), G ¨ung ¨or et al. ([17],[18]), Is¸ık [19], Mursaleen et al. ([23],[24],[25]), Rath and Tripathy [28], Salat [30] and many others.
Theα − density of a subset E of N was defined by C¸olak [3]. Let α be a real number such that 0 < α ≤ 1. Theα − density of a subset E of N is defined by
δα(E)= lim n
1
nα |{k ≤ n : k ∈ E}| provided the limit exists,
where |{k ≤ n : k ∈ E}| denotes the number of elements of E not exceeding n.
If x= (xk) is a sequence such that xksatisfies property p (k) for all k except a set ofα−density zero, then
we say that xksatisfies p (k) for “almost all k according toα” and we abbreviate this by ”a.a.k (α)”.
2010 Mathematics Subject Classification. 40A05;40C05;46A45
Keywords. Lacunary sequence, Statistical convergence, Modulus function Received: 18 July 2013; Accepted: 20 November 2013
Communicated by Hari M. Srivastava
The order of statistical convergence of a sequence of numbers was given by Gadjiev and Orhan [15] and after then statistical convergence of orderα and strong p−Ces`aro summability of order α studied by C¸olak [3].
Let x= (xk) ∈ w and 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
nα|{k ≤ n : |xk−L| ≥ε}| = 0
i.e. for a.a.k (α) |xk−L|< ε for every ε > 0, in which case we say that x is statistically convergent of order α,
to L. In this case we write Sα− lim x
k= L [3].
By a lacunary sequence we mean an increasing integer sequence θ = (kr) such that k0 = 0 and hr =
(kr−kr−1) → ∞ as r → ∞. Throught this paper the intervals determined by θ will be denoted by Ir= (kr−1, kr]
and the ratio kr
kr−1 will be abbreviated by qr.
Subsequently lacunary sequences have been studied in ([6],[11],[13],[14],[21]).
2. Main Results
In this section we give the main results of the paper. In Theorem 2.10 we give the inclusion relations between the sets of Sαθ−statistically convergent sequences for different α’s and different θ’s. In Theorem 2.12
we give the relationships between strong Nθα p −summability and strong Nβθ p −summability for different θ’s. In Theorem 2.14 we give the relationship between strong Nβ
θ p −summability and Sαθ−statistical
convergence for different θ’s.
Definition 2.1[32] Letθ = (kr) be a lacunary sequence and 0< α ≤ 1 be given. The sequence x = (xk) ∈ w is
said to be Sαθ−statistically convergent (or lacunary statistically convergent sequence of orderα) if there is a real number L such that
lim
r→∞
1 hαr
|{k ∈ Ir: |xk−L| ≥ε}| = 0,
where Ir= (kr−1, kr] and hαr denote theαth power (hr)αof hr, that is hα = hαr
=
hα1, hα2, ..., hαr, ... . In this case
we write Sαθ− lim xk = L. The set of all Sαθ−statistically convergent sequences will be denoted by Sαθ. For
θ = (2r) we shall write Sαinstead of Sα
θand in the special caseα = 1 and θ = (2r) we shall write S instead of
Sαθ.
The lacunary statistical convergence of orderα is well defined for 0 < α ≤ 1, but it is not well defined forα > 1 in general. For this x = (xk) be defined as follows:
xk=(1, k = 2r0, k , 2r r= 1, 2, 3, ... then both lim r→∞ 1 hαr |{k ∈ Ir: |xk− 1| ≥ε}| ≤ lim r→∞ kr−kr−1 2hαr = lim r→∞ hr 2hαr = 0
and lim r→∞ 1 hαr |{k ∈ Ir: |xk− 0| ≥ε}| ≤ lim r→∞ kr−kr−1 2hαr = lim r→∞ hr 2hαr = 0
forα > 1, such that x = (xk) lacunary statistically convergence of orderα, both to 1 and 0, i.e., Sαθ− lim xk= 1
and Sαθ− lim xk= 0. But this is impossible.
Definition 2.2[32] Letθ = (kr) be a lacunary sequence,α ∈ (0, 1] be any real number and p be a positive real
number. A sequence x is said to be strongly Nαθ p −summable (or strongly Nθ p −summable of orderα)
if there is a real number L such that lim r→∞ 1 hαr P k∈Ir |xk−L|p= 0.
In this case we write Nαθ p − lim xk= L. The set of all strongly Nθ p −summable sequences of orderα will
be denoted by Nαθ p. In the special caseα = 1 we shall write Nθ p instead of Nαθ p and also in the special
caseθ = (2r) we shall write wαpinstead of Nαθ p. If L = 0, then we shall write wα
p,0instead of wαp. The set of
all strongly Nθ p −summable sequences of orderα, to 0 will be denoted by Nαθ,0 p.
Definition 2.3 Let 0< α ≤ 1 and θ = (kr) be a lacunary sequence. The sequence x is said to be an Sαθ−Cauchy
sequence if there is a subsequencenxk0(r)
o
of x such that k0
(r) ∈ Irfor each r, limrxk0(r)= L, and for every ε > 0
lim r 1 hαr n k ∈ Ir: xk−xk0(r) ≥ε o =0.
Theorem 2.4[32] Let 0< α ≤ 1 and x = (xk), y = yk be sequences of real numbers, then
(i) If Sαθ− lim xk= x0and c ∈ C, then Sαθ− lim (cxk)= cx0,
(ii) If Sαθ− lim xk= x0and Sαθ− lim yk= y0, then Sαθ− lim xk+ yk= x0+ y0.
The proofs of the following two theorems are obtained by using techniques Fridy ([12],Theorem 1) and Fridy and Orhan ([14],Theorem 2) respectively, therefore we give them without proofs.
Theorem 2.5Let 0< α ≤ 1, then the following statements are equivalent: (i) x is a statistically convergent sequence of orderα;
(ii) x is a statistically Cauchy sequence of orderα;
(iii) x is a sequence for which there is a convergent sequence y such that xk= yka.a.k (α) .
Theorem 2.6Let 0< α ≤ 1 and θ = (kr) be a lacunary sequence. The sequence x is Sαθ−convergent if and
only if x is an Sαθ−Cauchy sequence.
Theorem 2.7Let 0< α ≤ 1 and θ = (kr) be a lacunary sequence. If lim infrqr> 1, then wαp ⊂Nθα p
. Proof. If lim infrqr> 1 there exists δ > 0 such that 1 + δ ≤ qrfor all r ≥ 1. Then for x ∈ wαp,0, we write
τα r = 1 hαr X k∈Ir |xi|p = 1 hαr kr X i=1 |xi|p− 1 hαr kr−1 X i=1 |xi|p = kαr hαr 1 kαr kr X i=1 |xi|p −k α r−1 hαr 1 kαr−1 kr−1 X i=1 |xi|p .
Since hr= kr−kr−1, we have kαr hαr ≤ (1+ δ) α δα and kαr−1 hαr ≤ 1 δα. Hence x ∈ Nθ,0α p.
Theorem 2.8Let 0< α ≤ 1 and θ = (kr) be a lacunary sequence. If lim supr kkαr
r−1 < ∞, then Nθ p ⊂ w α p.
Proof. Let lim supr kr
kαr−1 < ∞, then there exists a constant M > 0 such that kr
kαr−1 < M for all r ≥ 1. Now let
x ∈ Nθ,0 p andε > 0, then we can find R > 0 and K > 0 such that supi≥Rτi< ε and τi< K for all i = 1, 2, ... .
Then if t is any integer with kr−1< t ≤ kr, where r > R, we can write
1 tα t X i=1 |xi|p≤ 1 kαr−1 kr X i=1 |xi|p= 1 kαr−1 X I1 |xi|p+X I2 |xi|p+ ... +X Ir |xi|p = k1 kαr−1τ1+ k2−k1 kαr−1 τ2+ ... + kR−kR−1 kαr−1 τR+ kR+1−kR kαr−1 τR+1+ ... + kr−kr−1 kαr−1 τr ≤ sup i≥1 τi ! kR kαr−1 + supi≥R τi ! kr−kR kαr−1 < K kR kαr−1 + εM Hence x ∈ wαp,0. Theorem 2.9If x ∈ wα∩Nα
θand lim supr kr
kαr−1 < ∞ then Nαθ− lim xk= wα− lim xk.
Proof. Let Nαθ− lim xk = L and wα− lim xk = L 0
, and suppose that L , L0
. Since lim supr kr
kαr−1 < ∞ by
Theorem 2.8 we have Nθ,0 p ⊂ wαp,0. Since
x − L0∈ Nθ,0 p, it follows thatx − L0∈ wα p,0 and therefore 1 tα Pti=1 xi−L 0 → 0. Then we have 1 tα t X i=1 xi−L 0 + 1 tα t X i=1 |xi−L| ≥ 1 tα L−L 0 > 0, and hence L= L0.
Theorem 2.10Letθ = (kr) andθ0= (sr) be two lacunary sequences such that Ir⊂Jrfor all r ∈ N and let
α and β be such that 0 < α ≤ β ≤ 1, (i) If lim r→∞inf hαr `rβ > 0 (1) then Sβθ0 ⊆Sα θ , (ii) If lim r→∞ `r hβr = 1 (2) then Sαθ ⊆Sβ θ0.
Proof. (i)Suppose that Ir⊂Jrfor all r ∈ N and let (1) be satisfied. For given ε > 0 we have {k ∈ Jr: |xk−L| ≥ε} ⊇ {k ∈ Ir: |xk−L| ≥ε} and so 1 `rβ |{k ∈ Jr: |xk−L| ≥ε}| ≥h α r `βr 1 hαr |{k ∈ Ir: |xk−L| ≥ε}|
for all r ∈ N, where Ir = (kr−1, kr], Jr = (sr−1, sr], hr = kr−kr−1and `r = sr−sr−1. Now taking the limit as
r → ∞ in the last inequality and using (1) we get Sβθ0 ⊆Sα θ.
(ii) Let x= (xk) ∈ Sαθand (2) be satisfied. Since Ir⊂Jr, for ε > 0 we may write
1 `rβ |{k ∈ Jr: |xk−L| ≥ε}| = 1 `βr |{sr−1 < k ≤ kr−1 : |xk−L| ≥ε}| + 1 `βr |{kr< k ≤ sr: |xk−L| ≥ε}| + 1 `βr |{kr−1< k ≤ kr: |xk−L| ≥ε}| ≤ kr−1−sr−1 `βr +sr−kr `βr + 1 `βr |{k ∈ Ir: |xk−L| ≥ε}| = `r−hr `βr + 1 `rβ |{k ∈ Ir: |xk−L| ≥ε}| ≤ `r−h β r hβr + 1 hβr |{k ∈ Ir: |xk−L| ≥ε}| ≤ `r hβr − 1 + 1 hαr |{k ∈ Ir: |xk−L| ≥ε}|
for all r ∈ N. Since limr→∞ `r
hβr
= 1 by (2) the first term and since x = (xk) ∈ Sαθthe second term of right hand
side of above inequality tend to 0 as r → ∞ Note that `r hβr − 1 ≥ 0
. This implies that S
α θ⊆Sβθ0.
From Theorem 2.10 we have the following results.
Corollary 2.11Letθ = (kr) andθ0= (sr) be two lacunary sequences such that Ir⊆Jrfor all r ∈ N.
If (1) holds then, (i) Sαθ0 ⊆Sα θfor eachα ∈ (0, 1] , (ii) Sθ0 ⊆Sα θfor eachα ∈ (0, 1] , (iii) Sθ0 ⊆S θ. If (2) holds then, (i) Sαθ ⊆Sα θ0, for each α ∈ (0, 1] , (ii) Sαθ ⊆S θ0, for each α ∈ (0, 1] ,
(iii) Sθ⊆Sθ0.
Theorem 2.12Letθ = (kr) andθ0= (sr) be two lacunary sequences such that Ir⊆Jrfor all r ∈ N, α and β
be fixed real numbers such that 0< α ≤ β ≤ 1 and 0 < p < ∞. Then we have (i) If (1) holds then Nθβ0 p ⊂ Nα
θ p
, (ii) If (2) holds and x ∈`∞then Nαθ p ⊂ Nβ
θ0 p
. Proof. (i)Omitted.
(ii) Let x= (xk) ∈ Nαθ p and suppose that (2) holds. Since x= (xk) ∈`∞then there exists some M> 0
such that |xk−L| ≤ M for all k. Now, since Ir⊆Jrand hr≤`rfor all r ∈ N, we may write
1 `rβ X k∈Jr |xk−L|p= 1 `βr X k∈Jr−Ir |xk−L|p+ 1 `βr X k∈Ir |xk−L|p ≤ `r−hr `βr M p+ 1 `rβ X k∈Ir |xk−L|p ≤ `r−hβr hβr M p+ 1 hβr X k∈Ir |xk−L|p ≤ `r hβr − 1 M p+ 1 hαr X k∈Ir |xk−L|p
for every r ∈ N. Therefore x = (xk) ∈ Nβθ0 p
.
From Theorem 2.12 we have the following results.
Corollary 2.13Letθ = (kr) andθ0= (sr) be two lacunary sequences such that Ir⊂Jrfor all r ∈ N.
If (1) holds then, (i) Nαθ0 p ⊆ Nα θ p, for each α ∈ (0, 1] , (ii) Nθ0 p ⊆ Nα θ p, for each α ∈ (0, 1] , (iii) Nθ0 p ⊆ N θ p, If (2) holds then, (i)`∞∩Nθα p ⊂ Nα θ0 p , for each α ∈ (0, 1] , (ii)`∞∩Nθα p ⊂ Nθ0 p for eachα ∈ (0, 1] ,
(iii)`∞∩Nθ p ⊂ Nθ0 p.
Theorem 2.14Letθ = (kr) andθ0= (sr) be two lacunary sequences such that Ir⊆Jrfor all r ∈ N, α and β
be fixed real numbers such that 0< α ≤ β ≤ 1 and 0 < p < ∞. Then
(i) Let (1) holds, if a sequence is strongly Nβθ0 p −summable to L, then it is Sα
θ−statistically convergent
to L,
(ii) Let (2) holds, if a bounded sequence is Sαθ−statistically convergent to L then it is strongly Nθβ0 p −summable
Proof. (i)For any sequence x= (xk) andε > 0, we have X k∈Jr |xk−L|p= X k∈Jr |xk−L|≥ε |xk−L|p+ X k∈Jr |xk−L|<ε |xk−L|p ≥ X k∈Ir |xk−L|≥ε |xk−L|p ≥ |{k ∈ Ir: |xk−L| ≥ε}| εp and so that 1 `rβ X k∈Jr |xk−L|p≥ 1 `βr |{k ∈ Ir: |xk−L| ≥ε}| εp ≥ h α r `βr 1 hαr |{k ∈ Ir: |xk−L| ≥ε}| εp.
Since (1) holds it follows that if x = (xk) is strongly Nβθ0 p −summable to L, then it is Sα
θ−statistically
convergent to L.
(ii) Suppose that Sαθ− lim xk= L and x = (xk) ∈`∞. Then there exists some M > 0 such that |xk−L| ≤ M
for all k, then for every ε > 0 we may write 1 `rβ X k∈Jr |xk−L|p= 1 `βr X k∈Jr−Ir |xk−L|p+ 1 `βr X k∈Ir |xk−L|p ≤ `r−hr `βr M p+ 1 `rβ X k∈Ir |xk−L|p ≤ `r−hβr `rβ M p+ 1 `βr X k∈Ir |xk−L|p ≤ `r hβr − 1 M p+ 1 hβr X k∈Ir |xk−L|≥ε |xk−L|p+ 1 hβr X k∈Ir |xk−L|<ε |xk−L|p ≤ `r hβr − 1 M p+Mp hβr |{k ∈ Ir: |xk−L| ≥ε}| + hr hβr εp ≤ `r hβr − 1 M p+Mp hαr |{k ∈ Ir: |xk−L| ≥ε}| + `r hβr εp
for all r ∈ N. Using (2) we obtain that Nβθ0 p − lim xk= L, whenever Sα
θ− lim xk= L.
From Theorem 2.14 we have the following results.
Corollary 2.15Letα and β be fixed real numbers such that 0 < α ≤ β ≤ 1, 0 < p < ∞ and let θ = (kr) and
θ0= (s
r) be two lacunary sequences such that Ir⊂Jrfor all r ∈ N.
If (1) holds then,
(i) If a sequence is strongly Nαθ0 p −summable to L, then it is Sα
(ii) If a sequence is strongly Nθ0 p −summable to L, then it is Sα
θ−statistically convergent to L,
(iii) If a sequence is strongly Nθ0 p −summable to L, then it is S
θ−statistically convergent to L.
If (2) holds then,
(i) If a bounded sequence x= (xk) is Sαθ−statistically convergent to L then it is strongly Nαθ0 p −summable
to L,
(ii) If a bounded sequence x= (xk) is Sαθ−statistically convergent to L then it is strongly Nθ0 p −summable
to L,
(iii) If a bounded sequence x= (xk) is Sθ−statistically convergent to L then it is strongly Nθ0 p −summable
to L.
3. Results Related to Modulus Function
In this section we give the inclusion relations between the sets of Sαθ−statistically convergent sequences and strongly wα
(p)
θ, f −summable sequences with respect to the modulus function f.
The notion of a modulus was introduced by Nakano [26]. 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,
iv) f is continuous from the right at 0.
It follows that f must be continuous everywhere on [0, ∞). Maddox [22] and Ruckle [29] used a modulus function to construct some sequence spaces. Later on using a modulus different sequence spaces have been studied by Altin [1], Et ([7], [8]) ,Gaur and Mursaleen [16], Isık [20], Nuray and Savas [27] and many others. Definition 3.1Let f be a modulus function, p= pk be a sequence of strictly positive real numbers and
α ∈ (0, 1] be any real number. We define the sequence space wα
(p) θ, f as follows: wα(p)θ, f = ( x= (xk) : lim r→∞ 1 hαr P k∈Ir f (|xk−L|)pk = 0, for some L ) .
In the special case pk = p, for all k ∈ N and f (x) = x we shall write Nαθ p instead of wα
(p)
θ, f . If x ∈ wα
(p)
θ, f , then we say that x is strongly wα (p)
θ, f −summable with respect to the modulus function f and write wα
(p)
θ, f − lim x
k= L.
In the following theorems we shall assume that the sequence p= pk is bounded and 0< h = infkpk≤
pk≤ supkpk= H < ∞.
Theorem 3.2Letα, β ∈ (0, 1] be real numbers such that α ≤ β, f be a modulus function and let θ = (kr) be a
lacunary sequence, then wα
(p)
θ, f ⊂ Sβ
Proof.Let x ∈ wα (p)
θ, f and let ε > 0 be given and P
1and
P
2denote the sums over k ∈ Ir, |xk−L| ≥ε and
k ∈ Ir, |xk−L|< ε respectively. Since hαr ≤hβr for each r we may write
1 hαr P k∈Ir f (|xk−L|)pk = 1 hαr hP 1 f (|xk−L|)pk+ P2 f (|xk−L|)pk i ≥ 1 hβr hP 1 f (|xk−L|)pk+ P2 f (|xk−L|)pk i ≥ 1 hβr P 1 f (ε) pk ≥ 1 hβr P 1min( f (ε) h, f (ε)H ) ≥ 1 hβr |{k ∈ Ir: |xk−L| ≥ε}| min( f (ε)h, f (ε)H). Since x ∈ wα (p)
θ, f , the left hand side of the above inequality tends to zero as r → ∞. Therefore the right hand side tends to zero as r → ∞ and hence x ∈ Sβθ.
Theorem 3.3If the modulus f is bounded and limr→∞
hr
hαr
= 1 then Sα θ⊂wα(p)
θ, f .
Proof. Let x ∈ Sαθ and suppose that f is bounded andε > 0 be given. Since f is bounded there exists an integer K such that f (x) ≤ K, for all x ≥ 0. Then for each r ∈ N we may write
1 hαr P k∈Ir f (|xk−L|)pk = 1 hαr P 1 f (|xk−L|)pk + P2 f (|xk−L|)pk ≤ 1 hαr P 1max Kh, KH + 1 hαr P 2 f (ε) pk ≤ maxKh, KH 1 hαr k ∈ Ir: f (|xk−L|) ≥ε + hr hαr maxf(ε)h, f (ε)H . Hence x ∈ wα (p) θ, f .
Theorem 3.4If lim pk > 0 and x = (xk) is strongly wα
(p)
θ, f −summable to L with respect to the modulus function f , then wα
(p)
θ, f − lim x
k= L uniquely.
Proof.Let lim pk= s > 0. Suppose that wα(p)θ, f − lim xk= L, and wα(p)θ, f − lim xk= L1. Then
lim r 1 hαr P k∈Ir f (|xk−L|)pk = 0, and lim r 1 hαr P k∈Ir f (|xk−L1|)pk = 0. Definition of f , we have 1 hαr P k∈Ir f (|L − L1|)pk ≤ D hαr X k∈Ir f (|xk−L|)pk+ D hαr X k∈Ir f (|xk−L1|)pk,
where supkpk= H and D = max 1, 2H−1 . Hence lim r 1 hαr X k∈Ir f (|L − L1|)pk = 0. Since lim
k→∞pk= s we have L − L1= 0. Thus the limit is unique.
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