R E S E A R C H Open Access
Double almost statistical convergence of order α
Ekrem Sava¸s*
*Correspondence:
ekremsavas@yahoo.com Department of Mathematics, Istanbul Commerce University, Üsküdar, Istanbul, Turkey
Abstract
The goal of this paper is to define and studyλ-double almost statistical convergence of orderα. Further some inclusion relations are examined. We also introduce a new sequence space by combining the double almost statistical convergence and an Orlicz function.
MSC: Primary 40B05; secondary 40C05
Keywords: statistical convergence; Orlicz function; double statistical convergence of orderα; double almost statistical convergence
1 Introduction
The notion of statistical convergence was introduced by Fast [] and Schoenberg [] in- dependently. Over the years and under different names, statistical convergence was dis- cussed in the theory of Fourier analysis, ergodic theory and number theory. Later on it was further investigated from the sequence space point of view and linked with summa- bility theory by Fridy [], Connor [], ˘Salát [], Cakalli [], Miller [], Maddox [] and many others. However, Mursaleen [] defined the concept ofλ-statistical convergence as a new method and found its relation to statistical convergence, (C, )-summability and strong (V ,λ)-summability. Recently, for α ∈ (, ], Çolak and Bektaş [] have introduced theλ-statistical convergence of order α and strong (V, λ)-summability of order α for se- quences of complex numbers.
In this paper we define and studyλ-double almost statistical convergence of order α.
Also, some inclusion relations have been examined.
The notion of statistical convergence depends on the density of subsets ofN. A subset E ofN is said to have density δ(E) if
δ(E) = limn→∞ n
n k=
χE(k) exists.
Note that if K⊂ N is a finite set, then δ(K) = , and for any set K ⊂ N, δ(KC) = –δ(K).
Definition . A sequence x = (xk) is said to be statistically convergent to if for every ε > ,
δ
k∈ N : |xk–| ≥ ε
= .
We write st-lim xk= L in case x = (xk) is st-statistically convergent to L.
© 2013 Sava¸s; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu- tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Let wbe the set of all real or complex double sequences. By the convergence of a double sequence we mean the convergence in the Pringsheim sense, that is, the double sequence x = (xij)∞i,j=has a Pringsheim limit L denoted by P-lim x = L provided that, given > , there exists N∈ N such that |xij– L| < whenever i, j ≥ N. We will describe such an x more briefly as ‘P-convergent’ (see []).
We denote by c the space of P-convergent sequences. A double sequence x = (xij) is bounded ifx = supi,j≥|xij| < ∞. Let l∞ and c∞ be the set of all real or complex bounded double sequences and the set of bounded and convergent double sequences, respectively.
Móricz and Rhoades [] defined the almost convergence of the double sequence as fol- lows: x = (xij) is said to be almost convergent to a number L if
P- lim
p,q→∞sup
m,n
(p + )(q + )
m+p i=m
n+q j=n
xij– L
= ,
that is, the average value of (xi,j) taken over any rectangle
D =
(i, j) : m≤ i ≤ m + p, n ≤ j ≤ n + q
tends to L as both p and q tend to∞, and this convergence is uniform in m and n. We denote the space of almost convergent double sequences byˆcas
ˆc=
x = (xi,j) : lim
k,l→∞tklpq(x) – L= , uniformly in p, q
,
where
tklpq(x) = (k + )(l + )
k+p i=p
l+q j=q
xi,j.
The notion of almost convergence for single sequences was introduced by Lorentz []
and some others.
A double sequence x is called strongly double almost convergent to a number L if
P- lim
k,l→∞
(k + )(l + )
k+p i=p
l+q j=q
|xi,j– L| = uniformly in p, q.
By [ˆc] we denote the space of strongly almost convergent double sequences. It is easy to see that the inclusions c∞ ⊂ [ˆc]⊂ ˆc⊂ l∞strictly hold.
The notion of strong almost convergence for single sequences has been introduced by Maddox [].
A linear functional L on l∞ is said to be a Banach limit if it has the following properties:
() L(x) ≥ if x ≥ (i.e., xi,j≥ for all i, j),
() L(e) = , where e = (ei,j) with ei,j= for all i, j and
() L(x) = L(Sx) = L(Sx) = L(Sx), where the shift operators Sx, Sx, Sx are defined by Sx = (xi+,j), Sx = (xi,j+), Sx = (xi+,j+).
Let Bbe the set of all Banach limits on l∞. A double sequence x = (xi,j) is said to be almost convergent to a number L if L(x) = L for all L∈ B(see []).
The idea of statistical convergence was extended to double sequences by Mursaleen and Edely []. More recent developments on double sequences can be found in [–], where some more references can be found. For the single sequences, statistical convergence of orderα and strong p-Cesàro summability of order α was introduced by Çolak []. Quite recently, in [], Çolak and Bektaş generalized this notion by using de la Valée-Poussin mean.
Let K⊆ N × N be a two-dimensional set of positive integers and let Km,nbe the numbers of (i, j) in K such that i≤ n and j ≤ m.
Then the lower asymptotic density of K is defined as
P-lim inf
m,n
Km,n
mn =δ(K ).
In the case when the sequence (Kmnm,n)∞,∞m,n=,has a limit, we say that K has a natural density and is defined as
P-lim
m,n
Km,n
mn =δ(K ).
For example, let K ={(i, j) : i, j∈ N}, where N is the set of natural numbers. Then
δ(K ) = P-lim
m,n
Km,n
mn ≤ P-lim
m,n
√m√ n mn = (i.e., the set K has double natural density zero).
Mursaleen and Edely [] presented the notion statistical convergence for a double se- quence x = (xij) as follows: A real double sequence x = (xij) is said to be statistically con- vergent to L provided that for each > ,
P-lim
m,n
mn(i, j) : i≤ m and j ≤ n, |xij– L| ≥ = .
We now give the following definition.
The double statistical convergence of order α is defined as follows. Let < α ≤ be given. The sequence (xij) is said to be statistically convergent of orderα if there is a real number L such that
P- lim
mn→∞
(mn)αi≤ m and j ≤ n : |xij– L| ≥ =
for every > , in which case we say that x is double statistically convergent of order α to L.
In this case, we write Sα-lim xij= L. The set of all double statistically convergent sequences of orderα will be denoted by Sα. If we takeα = in this definition, we can have previous definition.
Letλ = (λn) be a non-decreasing sequence of positive numbers tending to∞ such that λn+≤ λn+ , λ= .
The generalized de la Valèe-Poussin mean is defined by
tn(x) = λn
k∈In
xk,
where In= [n –λn+ , n]. A sequence x = (xn) is said to be (V ,λ)-summable to a number L if tn(x)→ L as n → ∞.
In [] Mursaleen introduced the idea ofλ-statistical convergence for a single sequence as follows:
The number sequence x = (xi) is said to beλ-statistically convergent to the number if for each > ,
limn
λn
n –λn+ ≤ i ≤ n : |xi– L| ≥ = .
In this case, we write Sλ-limixi= and we denote the set of all λ-statistically convergent sequences by Sλ.
Definition . Let λ = (λm) andμ = (μn) be two non-decreasing sequences of positive real numbers both of which tend to ∞ as m and n approach ∞, respectively. Also, let λm+≤ λm+ ,λ= and μn+≤ μn+ ,μ= . We write the generalized double de la Valèe-Poussin mean by
tmn(x) = λmμn
i∈Im,j∈In
xij.
A sequence x = (xij) is said to be (V,λ, μ)-summable to a number L, if tmn(x)→ L as m, n→ ∞ in the Pringsheim sense. Throughout this paper, we denote ¯λmnbyλmμnand (i∈ Im, j∈ In) by (i, j)∈ Imn.
2 Main results
In this section, we defineλ-double almost statistically convergent sequences of order α.
Also, we prove some inclusion theorems.
We now have the following.
Definition . Let < α ≤ be given. The sequence x = (xij)∈ w is said to be ˆSα¯λ- statistically convergent of orderα if there is a real number L such that
P- lim
mn→∞
¯λαmn(k, l)∈ Imn:tklpq(x) – L ≥= uniformly in p, q,
where ¯λαmndenotes the αth power (¯λmn)α of ¯λmn. In case x = (xij) is ˆSα¯λ-statistically con- vergent of orderα to L, we write ˆSα¯λ-lim xij= L. We denote the set of all ˆSα¯λ-statistically convergent sequences of orderα by ˆSα¯λ. We write ˆSif ¯λmn= mn andα = for ˆSα¯λ.
We know that the ˆSα¯λ-statistical convergence of orderα is well defined for < α ≤ , but it is not well defined forα > in general. For this let x = (xij) be fixed. Then, for an arbitrary number L and > , we write
P- lim
mn→∞
¯λαmn
(k, l)∈ Imn:tklpq(x) – L ≥
≤ limn→∞[¯λmn] +
¯λαmn
= uniformly in p, q.
Therefore ˆSα¯λ-lim xijis not uniquely determined forα > .
Definition . Let < α ≤ be any real number and let r be a positive real number.
A sequence x is said to be strongly ˆwαr(¯λ)-summable of order α, if there is a real number L such that
P- lim
mn→∞
¯λαmn
(k,l)∈Imn
tklpq(x) – Lr= uniformly in p, q.
If we take α = , the strong ˆwαr(¯λ)-summability of order α reduces to the strong ˆwr(¯λ)- summability.
We denote the set of all strongly ˆwαr(¯λ)-summable sequences of order α by ˆwαp(¯λ).
We now are ready to state the following theorem.
Theorem . If < α ≤ β ≤ , then ˆSα¯λ⊂ ˆSβ¯λ.
Proof Let <α ≤ β ≤ . Then
¯λβmn
(k, l)∈ Imn:tklpq(x) – L ≥≤ ¯λαmn(k, l)∈ Imn:tklpq(x) – L ≥
for every > , and finally we have that ˆSα¯λ⊂ ˆSβ¯λ. This proves the result.
We have the following from the previous theorem.
Corollary .
(i) If a sequence is ˆSα¯λ-statistically convergent of orderα to L, then it is ˆS¯λ-statistically convergent to L, that is, ˆSα¯λ⊂ ˆS¯λfor eachα ∈ (, ],
(ii) α = β =⇒ ˆSα¯λ= ˆSβ¯λ, (iii) α = =⇒ ˆSα¯λ= ˆS¯λ. Theorem . ˆSα⊆ ˆSα¯λif
mn→∞lim inf ¯λαmn
(mn)α> . (.)
Proof For given > , we write
k≤ m and l ≤ m :tklpq(x) – L ≥⊃(k, l)∈ Imn:tklpq(x) – L ≥
and so
(mn)αk≤ m and l ≤ m :tklpq(x) – L ≥
≥ ¯λαmn
(mn)α
¯λαmn
(k, l)∈ Imn:tklpq(x) – L ≥.
Using (.) and taking the limit as mn→ ∞, we have ˆSα-lim xij= L =⇒ ˆSα¯λ-lim xkl= L.
Theorem . Let < α ≤ β ≤ and r be a positive real number, then ˆwαr(¯λ) ⊆ ˆwβr(¯λ).
Proof Let x = (xij)∈ ˆwαr(¯λ). Then given α and β such that < α ≤ β ≤ and a positive real number r, we write
¯λβmn
(k,l)∈Imn
tklpq(x) – Lr
and we get that ˆwαr(¯λ) ⊆ ˆwβr(¯λ).
We have the following corollary which is a consequence of Theorem ..
Corollary . Let < α ≤ β ≤ and r be a positive real number. Then (i) If α = β, then ˆwαr(¯λ) = ˆwβr(λ).
(ii) ˆwαr(¯λ) ⊆ ˆwr(¯λ) for each α ∈ (, ] and < r < ∞.
Theorem . Let α and β be fixed real numbers such that < α ≤ β ≤ and < r < ∞. If a sequence is a strongly ˆwαr(¯λ)-summable sequence of order α to L, then it is ˆSβ¯λ-statistically convergent of orderβ to L, i.e., ˆwαr(¯λ) ⊂ ˆSβ¯λ.
Proof For any sequence x = (xij) and > , we write
(k,l)∈Imn
tklpq(x) – Lr =
(k,l)∈Imn
|tklpq(x)–L|≥
tklpq(x) – Lr+
(k,l)∈Imn
|tklpq(x)–L|<
tklpq(x) – Lr
≥
(k,l)∈Imn
|tklpq(x)–L|≥
tklpq(x) – Lp≥(k, l)∈ Imn:tklpq(x) – L ≥·r
and so that
¯λαmn
(k,l)∈Imn
tklpq(x) – Lr≥
¯λαmn(k, l)∈ Imn:tklpq(x) – L ≥·r
≥
¯λβmn
(k, l)∈ Imn:tklpq(x) – L ≥·r.
This shows that if x = (xij) is a strongly ˆwαr(¯λ)-summable sequence of order α to L, then it is ˆSβ¯λ-statistically convergent of orderβ to L. This completes the proof.
We have the following corollary.
Corollary . Let α be fixed real numbers such that < α ≤ and < r < ∞.
(i) If a sequence is a strongly ˆwαr(¯λ)-summable sequence of order α to L, then it is ˆSα¯λ-statistically convergent of orderα to L, i.e., ˆwαr(¯λ) ⊂ ˆSα¯λ.
(ii) ˆwαr(¯λ) ⊂ ˆS¯λfor <α ≤ .
3 Some sequence spaces
In present section, we study the inclusion relations between the set of ˆSα¯λ-statistically con- vergent sequences of orderα and strongly ˆwαr[¯λ, M]-summable sequences of order α with respect to an Orlicz function M.
Recall in [] that an Orlicz function M : [,∞) → [, ∞) is continuous, convex, non- decreasing function such that M() = and M(x) > for x > , and M(x)→ ∞ as x → ∞.
An Orlicz function M is said to satisfy-condition for all values of u if there exists K > such that M(u)≤ KM(u), u ≥ .
Lindenstrauss and Tzafriri [] used the idea of an Orlicz function to construct the se- quence space
M=
x∈ w :
∞ k=
M |xk|
ρ
<∞ for some ρ >
.
The spaceMwith the norm
x = inf
ρ > :
∞ k=
M |xk|
ρ
≤
becomes a Banach space called an Orlicz sequence space. The spaceMis closely related to the spacepwhich is an Orlicz sequence space with M(x) =|x|rfor ≤ r < ∞.
In the later stage, different classes of Orlicz sequence spaces were introduced and studied by Parashar and Choudhary [], Savaş [–] and many others.
Definition . Let M be an Orlicz function, r = (rkl) be a sequence of strictly positive real numbers and letα ∈ (, ] be any real number. Now we write
ˆwαr[¯λ, M] =
x = (xkl) : P- lim
mn→∞
¯λαmn
(k,l)∈Imn
M(|tklpq(x) – L|) ρ
rkl
=
uniformly in p, q for some L andρ >
.
If x∈ ˆwαr[¯λ, M], then we say that x is almost strongly double λ-summable of order α with respect to the Orlicz function M.
If we consider various assignments of M, ¯λ and r in the above sequence spaces, we are granted the following:
() If M(x) = x, ¯λmn= mn and rk,l= for all (k, l), then ˆwαr[¯λ, M] = [ ˆwα].
() If rk,l= for all (k, l), then ˆwαr[¯λ, M] = ˆwα[¯λ, M].
() If rk,l= for all (k, l) and ¯λmn= mn, then ˆwαr[¯λ, M] = ˆwα[M].
() If ¯λmn= mn, then ˆwαr[¯λ, M] = ˆwαr[M].
We now have the following theorem.
Theorem . If rk,l> and x is almost stronglyλ-double convergent to Lwith respect to the Orlicz function M, that is, xkl→ L(ˆwαr[¯λ, M]), then Lis unique.
The proof of Theorem . is straightforward. So, we omit it.
In the following theorems, we assume that r = (rkl) is bounded and < h = infklrkl≤ rkl≤ supklrkl= H <∞.
Theorem . Let α, β ∈ (, ] be real numbers such that α ≤ β and M be an Orlicz func- tion, then ˆwαr[¯λ, M] ⊂ ˆSβ¯λ.
Proof Let x∈ ˆwαr[¯λ, M], > be given and
and
denote the sums over (k, l)∈ Imn,
|tklpq(x) – L| ≥ and (k, l) ∈ Imn,|tklpq(x) – L| < , respectively. Since λαmn≤ ¯λβmn, for each m, n we write
¯λαmn
(k,l)∈Imn
M(|tklpq(x) – L|) ρ
rkl
=
¯λαmn
M(|tklpq(x) – L|) ρ
rkl
+
M(|tklpq(x) – L|) ρ
rkl
≥
¯λβmn
M(|tklpq(x) – L|) ρ
rkl
+
M(|tklpq(x) – L|) ρ
rkl
≥
¯λβmn
f ()rkl
≥
¯λβmn
min
M()h
,
M()H
, = ρ
≥
¯λβmn
(k, l)∈ Imn:tklpq(x) – L ≥min
M()h
,
M()H .
Since x∈ ˆwαr[¯λ, M], the left-hand side of the above inequality tends to zero as mn → ∞ uniformly in p, q. Hence the right-hand side tends to zero as mn→ ∞ uniformly in p, q
and therefore x∈ ˆSβ¯λ. This proves the result.
Corollary . Let α ∈ (, ] and M be an Orlicz function, then ˆwαr[¯λ, M] ⊂ ˆSα¯λ.
We conclude this paper with the following theorem.
Theorem . Let M be an Orlicz function and x = (xij) be a bounded sequence, then ˆSα¯λ⊂ ˆwαr[¯λ, M].
Proof Suppose that x∈ ∞ and ˆSα¯λ-lim xij= L. Since x∈ ∞ , then there is a constant T > such that|tklpq(x)| ≤ T. Given > , we write for all p, q
¯λαmn
(k,l)∈Imn
M
|tklpq(x) – L| ρ
rkl
=
¯λαmn
M
|tklpq(x) – L| ρ
rkl
+
¯λαmn
M
|tklpq(x) – L| ρ
rkl
≤
¯λαmn
max
M T
ρ h
,
M
T ρ
H +
¯λαmn
M
ρ
rkl
≤ max
M(K )h
,
M(K )H
¯λαmn
(k, l)∈ Imn:tklpq(x) – L ≥
+ max
M()h
,
M()H , T
ρ = K , ρ =.
Therefore ˆwαp[¯λ, M]. This proves the result.
Competing interests
The author declares that they have no competing interests.
Received: 16 November 2012 Accepted: 15 February 2013 Published: 20 March 2013 References
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doi:10.1186/1687-1847-2013-62
Cite this article as: Sava¸s: Double almost statistical convergence of orderα. Advances in Difference Equations 2013 2013:62.
