R E S E A R C H
Open Access
Double almost statistical convergence of
order
α
Ekrem Sava¸s
**Correspondence:
[email protected] Department of Mathematics, Istanbul Commerce University, Üsküdar, Istanbul, Turkey
Abstract
The goal of this paper is to define and study
λ-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 order
α
; double almost statistical convergence1 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, forα∈(, ], Çolak and Bektaş [] have introduced theλ-statistical convergence of orderαand strong (V,λ)-summability of orderαfor se-quences of complex numbers.
In this paper we define and studyλ-double almost statistical convergence of orderα. Also, some inclusion relations have been examined.
The notion of statistical convergence depends on the density of subsets ofN. A subset EofNis said to have densityδ(E) if
δ(E) = lim
n→∞
n
n
k=
χE(k) exists.
Note that ifK⊂Nis a finite set, thenδ(K) = , and for any setK⊂N,δ(KC) = –δ(K).
Definition . A sequencex= (xk) is said to bestatistically convergenttoif for every
ε> ,
δk∈N:|xk–| ≥ε
= .
We writest-limxk=Lin casex= (xk) isst-statistically convergent toL.
Letwbe 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 limitLdenoted byP-limx=Lprovided that, given> ,
there exists N∈Nsuch that|xij–L|< wheneveri,j≥N. We will describe such anx
more briefly as ‘P-convergent’ (see []).
We denote byc the space ofP-convergent sequences. A double sequencex= (xij) is
bounded ifx=supi,j≥|xij|<∞. Letl∞ andc∞ 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 numberLif
P- lim
p,q→∞supm,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 toLas bothpandqtend to∞, and this convergence is uniform inmandn. We denote the space of almost convergent double sequences bycˆas
ˆ c=
x= (xi,j) : lim k,l→∞
tklpq(x) –L= , uniformly inp,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 sequencexis calledstrongly double almost convergentto a numberLif
P- lim
k,l→∞
(k+ )(l+ )
k+p
i=p l+q
j=q
|xi,j–L|= uniformly inp,q.
By [ˆc] we denote the space of strongly almost convergent double sequences. It is easy
to see that the inclusionsc∞ ⊂[cˆ]⊂ ˆc⊂l∞strictly hold.
The notion of strong almost convergence for single sequences has been introduced by Maddox [].
A linear functionalLonl∞ is said to be a Banach limit if it has the following properties:
() L(x)≥ifx≥(i.e.,xi,j≥for alli,j),
() L(e) = , wheree= (ei,j)withei,j= for alli,jand
() L(x) =L(Sx) =L(Sx) =L(Sx), where the shift operatorsSx,Sx,Sxare
defined bySx= (xi+,j),Sx= (xi,j+),Sx= (xi+,j+).
LetBbe the set of all Banach limits onl∞. A double sequencex= (xi,j) is said to be
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 orderαand strongp-Cesàro summability of orderαwas introduced by Çolak []. Quite recently, in [], Çolak and Bektaş generalized this notion by using de la Valée-Poussin mean.
LetK⊆N×Nbe a two-dimensional set of positive integers and letKm,nbe the numbers
of (i,j) inKsuch thati≤nandj≤m.
Then the lower asymptotic density ofKis defined as
P-lim inf
m,n
Km,n
mn =δ(K).
In the case when the sequence (Km,n
mn)
∞,∞
m,n=,has a limit, we say thatKhas a natural density
and is defined as
P-lim
m,n
Km,n
mn =δ(K).
For example, letK={(i,j) :i,j∈N}, whereNis the set of natural numbers. Then
δ(K) =P-lim
m,n
Km,n
mn ≤P-limm,n
√ m√n mn =
(i.e., the setKhas double natural density zero).
Mursaleen and Edely [] presented the notion statistical convergence for a double se-quencex= (xij) as follows: A real double sequencex= (xij) is said to be statistically
con-vergent toLprovided that for each> ,
P-lim
m,n
mn(i,j) :i≤mandj≤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
numberLsuch that
P- lim
mn→∞
(mn)αi≤mandj≤n:|xij–L| ≥=
for every> , in which case we say thatxis double statistically convergent of orderαtoL. In this case, we writeSα
-limxij=L. The set of all double statistically convergent sequences
of orderαwill be denoted bySα
. 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
whereIn= [n–λn+ ,n]. A sequencex= (xn) is said to be (V,λ)-summable to a numberL
iftn(x)→Lasn→ ∞.
In [] Mursaleen introduced the idea ofλ-statistical convergence for a single sequence as follows:
The number sequencex= (xi) is said to beλ-statistically convergent to the numberif
for each> ,
lim
n
λn
n–λn+ ≤i≤n:|xi–L| ≥= .
In this case, we writeSλ-limixi=and we denote the set of allλ-statistically convergent
sequences bySλ.
Definition . Letλ= (λm) andμ= (μn) be two non-decreasing sequences of positive
real numbers both of which tend to ∞asmandnapproach∞, 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 sequencex= (xij) is said to be (V,λ,μ)-summable to a numberL, iftmn(x)→Las
m,n→ ∞in the Pringsheim sense. Throughout this paper, we denoteλ¯mnbyλmμnand
(i∈Im,j∈In) by (i,j)∈Imn.
2 Main results
In this section, we defineλ-double almost statistically convergent sequences of orderα. 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 Sˆαλ¯ -statistically convergent of orderαif there is a real numberLsuch that
P- lim
mn→∞ ¯
λα
mn
(k,l)∈Imn:tklpq(x) –L≥= uniformly inp,q,
whereλ¯αmndenotes the αth power (¯λmn)α ofλ¯mn. In casex= (xij) isSˆλα¯-statistically con-vergent of orderα toL, we writeSˆα
¯
λ-limxij=L. We denote the set of allSˆ α ¯
λ-statistically convergent sequences of orderαbySˆα
¯
λ. We writeSˆifλ¯mn=mnandα= forSˆ α ¯ λ.
We know that theSˆα ¯
λ-statistical convergence of orderαis well defined for <α≤, but it is not well defined forα> in general. For this letx= (xij) be fixed. Then, for an arbitrary
numberLand> , we write
P- lim
mn→∞
¯
λα
mn
(k,l)∈Imn:tklpq(x) –L≥
≤ lim
n→∞
[¯λmn] +
¯
λα
mn
= uniformly inp,q.
ThereforeSˆα ¯
Definition . Let <α≤ be any real number and let rbe a positive real number. A sequencexis said to be stronglywˆα
r(λ¯)-summable of orderα, if there is a real numberL
such that
P- lim
mn→∞
¯
λα
mn
(k,l)∈Imn
tklpq(x) –Lr= uniformly inp,q.
If we take α= , the strongwˆα
r(λ¯)-summability of orderα reduces to the strongwˆr(λ¯
)-summability.
We denote the set of all stronglywˆα
r(λ¯)-summable sequences of orderαbywˆαp(λ¯).
We now are ready to state the following theorem.
Theorem . If <α≤β≤,thenSˆλα¯ ⊂ ˆSλβ¯.
Proof Let <α≤β≤. Then
¯
λβmn
(k,l)∈Imn:tklpq(x) –L≥≤
¯
λα
mn
(k,l)∈Imn:tklpq(x) –L≥
for every> , and finally we have thatSˆα ¯ λ⊂ ˆS
β ¯
λ. This proves the result. We have the following from the previous theorem.
Corollary .
(i) If a sequence isSˆα ¯
λ-statistically convergent of orderαtoL,then it isSˆ¯λ-statistically convergent toL,that is,Sˆα
¯
λ⊂ ˆSλ¯ for eachα∈(, ],
(ii) α=β =⇒ ˆSλ¯α=Sˆβλ¯, (iii) α= =⇒ ˆSλα¯ =Sˆλ¯.
Theorem . Sˆα
⊆ ˆSαλ¯ if
lim
mn→∞inf ¯
λαmn
(mn)α> . (.)
Proof For given> , we write
k≤mandl≤m:tklpq(x) –L≥
⊃(k,l)∈Imn:tklpq(x) –L≥
and so
(mn)αk≤mandl≤m:tklpq(x) –L≥ ≥ λ¯αmn
(mn)α ¯
λα
mn
(k,l)∈Imn:tklpq(x) –L≥.
Using (.) and taking the limit as mn→ ∞, we haveSˆα
-limxij=L =⇒ ˆSαλ¯-limxkl=L.
Theorem . Let <α≤β≤and r be a positive real number,thenwˆα
r(λ¯)⊆ ˆw
β
Proof Letx= (xij)∈ ˆwαr(λ¯). Then givenαandβsuch that <α≤β≤ and a positive real
numberr, we write
¯
λβmn
(k,l)∈Imn
tklpq(x) –Lr
and we get thatwˆα
r(λ¯)⊆ ˆw
β
r(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) Ifα=β,thenwˆα
r(¯λ) =wˆ
β
r(λ).
(ii) wˆα
r(¯λ)⊆ ˆwr(¯λ)for eachα∈(, ]and <r<∞.
Theorem . Letαandβbe fixed real numbers such that <α≤β≤and <r<∞.If a sequence is a stronglywˆα
r(λ¯)-summable sequence of orderαto L,then it isSˆ
β ¯
λ-statistically convergent of orderβto L,i.e.,wˆα
r(¯λ)⊂ ˆS
β ¯ λ.
Proof For any sequencex= (xij) and> , we write
(k,l)∈Imn
tklpq(x) –L r
=
(k,l)∈Imn
|tklpq(x)–L|≥
tklpq(x) –L r
+
(k,l)∈Imn
|tklpq(x)–L|<
tklpq(x) –L r
≥
(k,l)∈Imn
|tklpq(x)–L|≥
tklpq(x) –L p
≥(k,l)∈Imn:tklpq(x) –L≥·r
and so that
¯
λα
mn
(k,l)∈Imn
tklpq(x) –Lr
≥ ¯
λα
mn
(k,l)∈Imn:tklpq(x) –L≥·r
≥ ¯
λβmn
(k,l)∈Imn:tklpq(x) –L≥·r.
This shows that ifx= (xij) is a stronglywˆαr(λ¯)-summable sequence of orderαtoL, then it
isSˆβλ¯-statistically convergent of orderβtoL. This completes the proof. We have the following corollary.
Corollary . Letαbe fixed real numbers such that <α≤and <r<∞.
(i) If a sequence is a stronglywˆα
r(¯λ)-summable sequence of orderαtoL,then it is
ˆ Sα
¯
λ-statistically convergent of orderαtoL,i.e.,wˆ α
r(λ¯)⊂ ˆSλα¯.
(ii) wˆα
r(λ¯)⊂ ˆSλ¯ for <α≤. 3 Some sequence spaces
In present section, we study the inclusion relations between the set ofSˆα ¯
λ-statistically con-vergent sequences of orderαand stronglywˆα
r[λ¯,M]-summable sequences of orderαwith
Recall in [] that an Orlicz functionM: [,∞)→[,∞) is continuous, convex, non-decreasing function such thatM() = andM(x) > forx> , andM(x)→ ∞asx→ ∞. An Orlicz functionMis said to satisfy -condition for all values ofuif there exists
K> such thatM(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
|x
k|
ρ
<∞for someρ>
.
The spaceMwith the norm
x=inf
ρ> : ∞
k=
M
|x
k|
ρ
≤
becomes a Banach space called an Orlicz sequence space. The spaceMis closely related
to the spacepwhich is an Orlicz sequence space withM(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 . LetMbe an Orlicz function,r= (rkl) be a sequence of strictly positive real
numbers and letα∈(, ] be any real number. Now we write
ˆ
wαr[λ¯,M] =
x= (xkl) :P- lim mn→∞
¯
λα
mn
(k,l)∈Imn
M(|tklpq(x) –L|)
ρ
rkl
=
uniformly inp,qfor someLandρ>
.
Ifx∈ ˆwα
r[¯λ,M], then we say thatxis almost strongly doubleλ-summable of orderαwith
respect to the Orlicz functionM.
If we consider various assignments ofM,λ¯andrin the above sequence spaces, we are granted the following:
() IfM(x) =x,λ¯mn=mnandrk,l= for all(k,l), thenwˆαr[λ¯,M] = [wˆα].
() Ifrk,l= for all(k,l), thenwˆαr[λ¯,M] =wˆα[λ¯,M].
() Ifrk,l= for all(k,l)andλ¯mn=mn, thenwˆαr[λ¯,M] =wˆα[M].
() Ifλ¯mn=mn, thenwˆαr[λ¯,M] =wˆαr[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(wˆαr[¯λ,M]),then Lis unique.
The proof of Theorem . is straightforward. So, we omit it.
In the following theorems, we assume thatr= (rkl) is bounded and <h=infklrkl≤
rkl≤supklrkl=H<∞.
Theorem . Letα,β∈(, ]be real numbers such thatα≤βand M be an Orlicz func-tion,thenwˆα
r[λ¯,M]⊂ ˆS
Proof Letx∈ ˆwα
uniformly inp,q. Hence the right-hand side tends to zero asmn→ ∞uniformly inp,q
and thereforex∈ ˆSβλ¯. This proves the result.
Corollary . Letα∈(, ]and M be an Orlicz function,thenwˆα
r[¯λ,M]⊂ ˆSαλ¯. We conclude this paper with the following theorem.
Competing interests
The author declares that they have no competing interests.
Received: 16 November 2012 Accepted: 15 February 2013 Published: 20 March 2013 References
1. Fast, H: Sur la convergence statistique. Colloq. Math.2, 241-244 (1951)
2. Schoenberg, IJ: The integrability of certain functions and related summability methods. Am. Math. Mon.66, 361-375 (1959)
3. Fridy, JA: On statistical convergence. Analysis5, 301-313 (1985)
4. Connor, J: The statistical and strongp-Cesàro convergence of sequences. Analysis8, 47-63 (1988) 5. ˘Salát, T: On statistical convergence of real numbers. Math. Slovaca30, 139-150 (1980)
6. Cakalli, H: A study on statistical convergence. Funct. Anal. Approx. Comput.1(2), 19-24 (2009)
7. Miller, HI: A measure theoretical subsequence characterization of statistical convergence. Trans. Am. Math. Soc. 347(5), 1811-1819 (1995)
8. Maddox, IJ: On strong almost convergence. Math. Proc. Camb. Philos. Soc.85(2), 345-350 (1979) 9. Mursaleen, M:λ-statistical convergence. Math. Slovaca50(1), 111-115 (2000)
10. Çolak, R, Bekta¸s, CA:λ-statistical convergence of orderα. Acta Math. Sci., Ser. B31(3), 953-959 (2011) 11. Pringsheim, A: Zur theorie der zweifach unendlichen Zahlenfolgen. Math. Ann.53, 289-321 (1900)
12. Móricz, F, Rhoades, BE: Almost convergence of double sequences and strong regularity of summability matrices. Math. Proc. Camb. Philos. Soc.104, 283-294 (1988)
13. Lorentz, GG: A contribution to the theory of divergent sequences. Acta Math.80, 167-190 (1948)
14. Basarir, M, Konca, S: On some lacunary almost convergent double sequence spaces and Banach limits. Abstr. Appl. Anal.2012, Article ID 426357 (2012)
15. Mursaleen, M, Edely, OH: Statistical convergence of double sequences. J. Math. Anal. Appl.288(1), 223-231 (2003) 16. Bhunia, S, Das, P, Pal, S: Restricting statistical convergence. Acta Math. Hung.134(1-2), 153-161 (2012)
17. Cakalli, H, Sava¸s, E: Statistical convergence of double sequences in topological groups. J. Comput. Anal. Appl.12(2), 421-426 (2010)
18. Mursaleen, M, Çakan, C, Mohiuddine, SA, Sava¸s, E: Generalized statistical convergence and statistical core of double sequences. Acta Math. Sin. Engl. Ser.26(11), 2131-2144 (2010)
19. Çolak, R: Statistical convergence of orderα. In: Modern Methods in Analysis and Its Applications, pp. 121-129. Anamaya Pub., New Delhi (2010)
20. Krasnosel’skii, MA, Rutisky, YB: Convex Functions and Orlicz Spaces. Noordhoff, Groningen (1961) 21. Lindenstrauss, J, Tzafriri, L: On Orlicz sequence spaces. Isr. J. Math.101, 379-390 (1971)
22. Parashar, SD, Choudhary, B: Sequence space defined by Orlicz function. Indian J. Pure Appl. Math.25(14), 419-428 (1994)
23. Sava¸s, E: (A,λ)-double sequence spaces defined by Orlicz function and double statistical convergence. Comput. Math. Appl.55(6), 1293-1301 (2008)
24. Sava¸s, E, Sava¸s, R: Someλ-sequence spaces defined by Orlicz functions. Indian J. Pure Appl. Math.34(12), 1673-1680 (2003)
25. Sava¸s, E: On some new double lacunary sequences spaces via Orlicz function. J. Comput. Anal. Appl.11(3), 423-430 (2009)
26. Sava¸s, E, Sava¸s, R: Some sequence spaces defined by Orlicz functions. Arch. Math.40(1), 33-40 (2004)
27. Sava¸s, E, Patterson, RF: (λ,¯σ)-double sequence spaces via Orlicz function. J. Comput. Anal. Appl.10(1), 101-111 (2008) 28. Sava¸s, E, Patterson, RF: Double sequence spaces defined by Orlicz functions. Iran. J. Sci. Technol., Trans. A, Sci.31(2),
183-188 (2007)
doi:10.1186/1687-1847-2013-62
Cite this article as:Sava¸s:Double almost statistical convergence of orderα.Advances in Difference Equations2013
