On λ statistical convergence and strongly λ summable functions of order α

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R E S E A R C H

Open Access

On

λ-statistical convergence and strongly

λ-summable functions of order

α

Mikail Et

1

_{, Syed Abdul Mohiuddine}

2*

_{and Abdullah Alotaibi}

2

*_{Correspondence:} mohiuddine@gmail.com 2_{Department of Mathematics,} Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

Full list of author information is available at the end of the article

Abstract

In this study, using the notion of (V,

λ

)-summability and

λ

-statistical convergence, we introduce the concepts of strong (V,

λ

,p)-summability and

λ

-statistical convergence of order

α

of real-valued functions which are measurable (in the Lebesgue sense) in the interval (1,∞). Also some relations between

λ

-statistical convergence of order

α

and strong (V,

λ

,p)-summability of order

β

are given. MSC: 40A05; 40C05; 46A45

Keywords: statistical convergence; measurable function; (V,

λ

)-summability

1 Introduction

The idea of statistical convergence was given by Zygmund [] in the first edition of his monograph published in Warsaw in . The concept of statistical convergence was in-troduced by Steinhaus [] and Fast [] and then reinin-troduced by Schoenberg [] inde-pendently. Over the years and under different names, statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory, number theory, measure the-ory, trigonometric series, turnpike theory and Banach spaces. Later on it was further in-vestigated from the sequence space point of view and linked with summability theory by Alotaibi and Alroqi [], Belen and Mohiuddine [], Connor [], Duttaet al.[–], Etet al.[–], Fridy [], Güngöret al.[, ], Kolk [], Mohiuddineet al.[–], Mur-saleenet al.[–], Nuray [], Rath and Tripathy [], Salat [], Savaşet al.[, ], Tripathy [] and many others. In recent years, generalizations of statistical convergence have 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 are also connected with subsets of the Stone-Čech compactification of the natural numbers. Moreover, statistical convergence is closely related to the concept of convergence in probability.

2 Deﬁnition and preliminaries

The definitions of statistical convergence and strongp-Cesàro convergence of a sequence of real numbers were introduced in the literature independently of one another and fol-lowed different lines of development since their first appearance. It turns out, however, that the two definitions can be simply related to one another in general and are equiva-lent for bounded sequences. The idea of statistical convergence depends on the density of

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subsets of the setNof natural numbers. The density of a subsetEofNis deﬁned by

δ(E) =lim n→∞



n

k=

χE(k) provided the limit exists,

whereχEis the characteristic function ofE. It is clear that any ﬁnite subset ofNhas zero

natural density andδ(Ec_{) =  –}_δ₍_E_).

A sequencex= (xk) is said to be statistically convergent toLif for everyε> ,δ({k∈N:

|xk–L| ≥ε}) = . In this case we writexk

stat

−→LorS-limxk=L. The set of all statistically

convergent sequences will be denoted byS.

The order of statistical convergence of a sequence of numbers was given by Gadjiev

and Orhan in [] and after then statistical convergence of orderαand strongp-Cesàro

summability of order α were studied by Çolak [, ] and generalized by Çolak and

Bektaş [].

Letλ= (λn) be a non-decreasing sequence of positive real numbers tending to∞such

thatλn+≤λn+ ,λ= . The generalized de la Vallée-Poussin mean is deﬁned bytn(x) =

 λn

k∈Inxk, whereIn= [n–λn+ ,n] forn= , , . . . . A sequencex= (xk) is said to be (V,λ

)-summable to a numberLiftn(x)→Lasn→ ∞[]. Ifλn=n, then (V,λ)-summability is

reduced to Cesàro summability. Bywe denote the class of all non-decreasing sequence

of positive real numbers tending to∞such thatλn+≤λn+ ,λ= .

In [] Borwein introduced and studied strongly summable functions. A real-valued function x(t), measurable (in the Lebesgue sense) in the interval (,∞), is said to be strongly summable toL=Lxif

lim

n→∞



n



x(t) –Lpdt= , ≤p<∞.

[Wp] will denote the space of real-valued functionx, measurable (in the Lebesgue sense)

in the interval (,∞). The space [Wp] is a normed space with the norm

x=sup

n≥



n



x(t)pdt 

p

.

In this paper, using the notion of (V,λ)-summability andλ-statistical convergence, we introduce and study the concepts of strong (V,λ,p)-summability andλ-statistical conver-gence of orderαof real-valued functionsx(t), measurable (in the Lebesgue sense) in the interval (,∞).

Throughout the paper, unless stated otherwise, by ‘for alln∈Nno’ we mean ‘for alln∈

Nexcept ﬁnite numbers of positive integers’ whereNno={no,no+ ,no+ , . . .}for some

no∈N={, , , . . .}.

3 Main results

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the interval (,∞). In Theorem . we give the relationship between the strong [Wλβp

]-summability and [Sα

λ]-statistical convergence of real-valued functions which are measur-able (in the Lebesgue sense) in the interval (,∞).

Deﬁnition . Let the sequenceλ= (λn) be as above,α∈(, ] andx(t) be a real-valued

function which is measurable (in the Lebesgue sense) in the interval (,∞). A real-valued functionx(t) is said to be strongly (V,λ,p)-summable of orderα(or [Wα

λp]-summable) if

there is a numberL=Lxsuch that

lim

n→∞

 λα

n

n–λn+

x(t) –Lpdt= , ≤p<∞,

where In= [n–λn+ ,n] andλαn denote theαth power (λn)α ofλn, that is,λα= (λαn) =

(λα_,λα_, . . . ,λα

n, . . .). In this case we write [W

α

λp]-limx(t) =L. The set of all strongly (V,λ,p

)-summable functions of orderαwill be denoted by [Wλαp]. Forλn=nfor alln∈N, we shall

write [Wα

p] instead of [Wλαp] and in the special caseα=  we shall write [Wλp] instead of

[Wα

λp], and also in the special caseα=  andλn=nfor alln∈Nwe shall write [Wp] instead

of [Wα λp].

Deﬁnition . Let the sequenceλ= (λn) be as above,α∈(, ] andx(t) be a real-valued

function which is measurable (in the Lebesgue sense) in the interval (,∞). A real-valued functionx(t) is said to beλ-statistically convergent of orderα(or [Sλα]-statistical conver-gence) to a numberL=Lxfor everyε> ,

lim

n

 λα

n

t∈In:x(t) –L≥ε = .

The set of allλ-statistically convergent functions of orderαwill be denoted by [Sα λ]. In this case we write [Sα

λ]-limx(t) =L. Forλn=n, for alln∈N, we shall write [S

α_{] instead of [}_Sα λ] and in the special caseα= , we shall write [Sλ] instead of [Sα

λ].

Theλ-statistical convergence of orderαis well defined for  <α≤, but it is not well defined forα>  in general. For this letλn=nfor alln∈Nandx(t) be defined as follows:

x(t) =

 ift= n,

 ift= n, n= , , , . . . ,

then both

lim

n→∞

 λα

n

t∈In:x(t) – ≥ε ≤ lim n→∞

[λn] + 

λα

n

= 

and

lim

n→∞

 λα

n

t∈In:x(t) – ≥ε ≤ lim n→∞

([λn] + )

λα

n

= 

forα> , and soλ-statistically converges of orderαboth to  and ,i.e., [Sαλ]-limx(t) =  and [Sα

λ]-limx(t) = . But this is impossible.

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Theorem . Let <α≤and x(t)and y(t)be real-valued functions which are measur-able(in the Lebesgue sense)in the interval(,∞),then

(i) If[Sλα]-limx(t) =Landc∈R,then[S α

λ]-limcx(t) =cL; (ii) If[Sα

λ]-limx(t) =Land[Sαλ]-limy(t) =L,then[Sαλ]-lim(x(t) +y(t)) =L+L.

Theorem . Letλ= (λn)∈( <α≤β≤)and x(t)be a real-valued function which is

measurable(in the Lebesgue sense)in the interval(,∞),then[Sα λ]⊆[S

β λ].

From Theorem . we have the following.

Corollary . If x(t)isλ-statistically convergent of orderα to L,then it is statistically convergent to L for eachα∈(, ].

Theorem . Letλ= (λn)∈( <α≤)and x(t)be a real-valued function which is

measurable(in the Lebesgue sense)in the interval(,∞),then[Sα_]_⊆_[_Sα λ]if

lim

n→∞inf

λα

n

nα > . ()

Proof For givenε> , we have

t≤n:x(t) –L≥ε ⊃t∈In:x(t) –L≥ε ,

and so



nαt≤n:x(t) –L≥ε ≥ λα

n

nα  λα

n

t∈In:x(t) –L≥ε .

Taking the limit asn→ ∞and using (), we get [Sα_]-_limx

k=L⇒[Sαλ]-limxk=L.

From Theorem . we have the following result.

Corollary . Letλ= (λn)∈( <α≤)and x(t)be a real-valued function which is

measurable(in the Lebesgue sense)in the interval(,∞),then[S]⊆[Sα λ]if

lim

n→∞inf

λα_n

n > .

Theorem . Letλ= (λn)∈,  <α≤β≤,p be a positive real number and x(t)be a

real-valued function which is measurable(in the Lebesgue sense)in the interval(,∞),then

[Wα λp]⊆[W

β λp].

Proof Omitted.

From Theorem . we have the following result.

Corollary . Letλ= (λn)∈and p be a positive real number and x(t)be a real-valued

function which is measurable(in the Lebesgue sense)in the interval(,∞),then[Wα λp]⊆

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Theorem . Letλ= (λn)∈,  <α≤β≤,p be a positive real number and x(t)be a

real-valued function which is measurable(in the Lebesgue sense)in the interval(,∞).If a function x(t)is[Wα

λp]-summable,then it isλ-statistically convergent of orderβ.

Proof For any sequencex(t)∈[Wλαp] andε> , we have

n

n–λn+

x(t) –Lpdt=

n

n–λn+ |x(t)–L|≥ε

x(t) –Lpdt+

n

n–λn+ |x(t)–L|<ε

x(t) –Lpdt

≥

n

n–λn+ |x(t)–L|≥ε

x(t) –Lpdt

≥t∈In:x(t) –L≥ε ·εp

and so that

 λα

n

n–λn+

x(t) –Lpdt≥ 

λβn

t∈In:x(t) –L≥ε ·εp.

From this it follows that ifx(t) is [Wα

λp]-summable, then it isλ-statistically convergent of

orderβ.

From Theorem . we have the following results.

Corollary . Letαbe a ﬁxed real number such that <α≤and <p<∞.The fol-lowing statements hold:

(i) Ifx(t)is strongly[Wα

λp]-summable of orderα,then it isλ-statistically convergent of

orderα.

(ii) Ifx(t)is strongly[Wα

λp]-summable of orderα,then it isλ-statistically convergent.

Theorem . Letλ= (λn)andμ= (μn)be two sequences insuch thatλn≤μnfor all

n∈Nno( <α≤β≤),and let x(t)be a real-valued function which is measurable(in the Lebesgue sense)in the interval(,∞).If

lim inf

n→∞

λα

n

μβn

> , ()

then[Sβ

μ]⊆[Sαλ].

Proof Suppose thatλn≤μnfor alln∈Nnoand let () be satisﬁed. ThenIn⊂Jnand so that

ε>  we may write

t∈Jn:x(t) –L≥ε ⊃

t∈In:x(t) –L≥ε

and so



μβn

t∈Jn:x(t) –L≥ε ≥

λα_n μβn

 λα

n

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for alln∈Nno, whereJn= [n–μn+ ,n]. Now, taking the limit asn→ ∞in the last

in-equality and using (), we get [Sβ

μ]⊆[Sαλ].

From Theorem . we have the following results.

Corollary . Letλ= (λn)andμ= (μn)be two sequences insuch thatλn≤μnfor all

n∈Nno,and let x(t)be a real-valued function which is measurable(in the Lebesgue sense) in the interval(,∞).If()holds,then

(i) [Sα μ]⊆[S

α

λ]for eachα∈(, ], (ii) [Sμ]⊆[Sαλ]for eachα∈(, ], (iii) [Sμ]⊆[Sλ].

Theorem . Letλ= (λn)andμ= (μn)be two sequences insuch thatλn≤μnfor all

n∈Nno( <α≤β≤),and let x(t)be a real-valued function which is measurable(in the Lebesgue sense)in the interval(,∞).If()holds,then[Wμβp]⊆[Wλαp].

Proof Omitted.

Corollary . Letλ= (λn)andμ= (μn)be two sequences insuch thatλn≤μnfor all

(i) [Wα

μp]⊆[Wλαp]for eachα∈(, ],

(ii) [Wμp]⊆[Wλαp]for eachα∈(, ],

(iii) [Wμp]⊆[Wλp].

Theorem . Letλ= (λn)andμ= (μn)be two sequences insuch thatλn≤μnfor all

n∈Nno( <α≤β≤),and let x(t)be a real-valued function which is measurable(in the Lebesgue sense)in the interval(,∞)and()holds.If a real-valued function x(t)is strongly

(V,μ,p)-summable of orderβto L,then it isλ-statistically convergent of orderαto L.

Proof Letx(t) be a real-valued function such thatx(t) is strongly (V,μ,p)-summable of orderβtoLandε> . Then we have

t∈Jn

x(t) –Lpdt=

t∈Jn |x(t)–L|≥ε

x(t) –Lpdt+

t∈Jn |x(t)–L|<ε

x(t) –Lpdt

≥

t∈In |x(t)–L|≥ε

x(t) –Lpdt

≥t∈In:x(t) –L≥ε ·εp

and so that

 μβn

t∈Jn

x(t) –Lpdt≥ λ

α

n

μβn

 λα

n

t∈In:x(t) –L≥ε ·εp.

Since () holds, it follows that ifx(t) is strongly (V,μ,p)-summable of orderβtoL, then

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Corollary . Letλ= (λn)andμ= (μn)be two sequences insuch thatλn≤μnfor all

(i) A real-valued functionx(t)is strongly(V,μ,p)-summable of orderαtoL,then it is

λ-statistically convergent of orderαtoL;

(ii) A real-valued functionx(t)is strongly(V,μ,p)-summable toL,then it is

λ-statistically convergent of orderαtoL;

(iii) A real-valued functionx(t)is strongly(V,μ,p)-summable toL,then it is

λ-statistically convergent toL.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors contributed equally and signiﬁcantly in writing this paper. All the authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Firat University, Elazıg, 23119, Turkey.}2_{Department of Mathematics, Faculty of Science,} King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia.

Acknowledgements

The authors gratefully acknowledge the ﬁnancial support from King Abdulaziz University, Jeddah, Saudi Arabia.

Received: 22 March 2013 Accepted: 16 October 2013 Published:07 Nov 2013

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