R E S E A R C H
Open Access
On pointwise and uniform statistical
convergence of order
α
for sequences of
functions
Muhammed Çinar
1*, Murat Karaka¸s
2and Mikail Et
3*Correspondence:
[email protected] 1Department of Mathematics, Mu¸s
Alparslan University, Mu¸s, Turkey Full list of author information is available at the end of the article
Abstract
In this paper, we introduce the concepts of pointwise and uniform statistical
convergence of order
α
for sequences of real-valued functions. Furthermore, we give the concept of anα-statistically Cauchy sequence for sequences of real-valued
functions and prove that it is equivalent to pointwise statistical convergence of orderα
for sequences of real-valued functions. Also, some relations betweenSα(f)-statistical convergence and strongwβp(f)-summability are given.
MSC: 40A05; 40C05; 46A45
Keywords: statistical convergence; sequences of functions; Cesàro 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 later reinin-troduced by Schoenberg [] indepen-dently. Over the years and under different names, statistical convergence has been dis-cussed in the theory of Fourier analysis, ergodic theory, number theory, measure theory, trigonometric series, turnpike theory and Banach spaces. Later on it was further investi-gated from the sequence space point of view and linked with summability theory by Başar [], Connor [], Etet al.[–], Fridy [], Güngöret al.[], Işık [, ], Kolk [],
Mo-hiuddineet al. [–], Miller and Orhan [], Mursaleen [], Rath and Tripathy [],
Salat [], Savaş [] and many others. In recent years, generalizations of statistical con-vergence 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.
The definitions of pointwise and uniform statistical convergence of sequences of
real-valued functions were given by Gökhanet al. [, ] and independently by Duman and
Orhan []. In the present paper, we introduce and examine the concepts of pointwise
and uniform statistical convergence of orderα for sequences of real-valued functions.
In Section we give a brief overview of statistical convergence of orderαand strongp -Cesàro summability. In Section we give the concepts of pointwise and uniform statistical convergence of orderα, and the conceptα-statistically Cauchy sequence for sequences of
real-valued functions and prove that it is equivalent to pointwise statistical convergence of orderαfor sequences of real-valued functions. We also establish some inclusion relations betweenwβp(f) andSα(f) and betweenSα(f) andS(f).
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 have followed 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
subsets of the setNof natural numbers. The density of a subsetEofNis defined by
δ(E) =lim
n→∞
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).
Theα-densityof a subsetEofNwas defined by Çolak []. Letαbe a real number such that <α≤. Theα-densityof a subsetEofNis defined by
δα(E) =lim
n
nα{k≤n:k∈E} provided the limit exists,
where|{k≤n:k∈E}|denotes the number of elements ofEnot exceedingn.
Ifx= (xk) is a sequence such thatxksatisfies propertyP(k) for almost allkexcept a set
ofα-densityzero, then we say thatxksatisfies propertyP(k) for ‘almost all k according to
α’ and we abbreviate this by ‘a.a.k(α)’.
It is clear that any finite subset ofNhas zeroαdensity andδα(Ec) = –δα(E) does not hold for <α< in general, the equality holds only ifα= . Note that theα-densityof any set reduces to the natural density of the set in caseα= .
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 orderαwere studied by Çolak [].
The statistical convergence of orderαis defined as follows. Let <α≤ be given. The sequence (xk) is said to be statistically convergent of orderαif there is a real number
such that
lim
n→∞
nαk≤n:|xk–| ≥ε= ,
for everyε> , in which case we say thatxis statistically convergent of orderαto. In this case, we writeSα–limx
k=. The set of all statistically convergent sequences of order
αwill be denoted bySα. We writeSα
to denote the set of all statistically null sequences of orderα. It is clear thatSα
⊂Sαfor each <α≤. The statistical convergence of orderα is same with the statistical convergence forα= .
A sequencex= (xk) is said to be strongly Cesàro summable to a numberiflimnn×
n
defined as
[C, ] =
x= (xk) :lim n
n n
k=
|xk–|= for some
.
There is a natural relationship between statistical convergence and strong p-Cesàro
summability.
3 Main result
In this section we give the main results of this article. We give relations between the
sta-tistical convergence of orderα and the statistical convergence of orderβ for sequences
of functions, the relations between the strongp-Cesàro summability of orderαand the
strongp-Cesàro summability of orderβand the relations between the strongp-Cesàro
summability of orderαand the statistical convergence of orderβfor sequences of
real-valued functions, whereα≤β.
Definition . Let <α≤ be given. A sequence of functions{fk}is said to be pointwise
statistically convergent of orderα (or pointwiseα-statistically convergent sequence) to the functionf on a setAif, for everyε> ,
lim
n
nαk≤n:fk(x) –f(x)≥εfor everyx∈A=
i.e., for everyx∈A,
fk(x) –f(x)<ε a.a.k(α). ()
In this case, we writeSα–limf
k(x) =f(x) onA.Sα–limfk(x) =f(x) means that for every
δ> and <α≤, there is an integerNsuch that
nαk≤n:fk(x) –f(x)≥εfor everyx∈A<δ
for alln>N(=N(ε,δ,x)) and for eachε> . The set of all pointwise statistically convergent sequences of functions orderαwill be denoted bySα(f). Forα= , we will writeS(f) instead ofSα(f) and in the special casef = , we will writeSα
(f) instead ofSα(f).
The statistical convergence of orderα for a sequence of functions is well defined for
<α≤. But it is not well defined forα> . For this, let{fk}be defined as follows:
fk(x) = ⎧ ⎨ ⎩
, k= n,
xk, k= n, n= , , , . . . ,x∈
,
.
Then both
lim
n→∞
nαk≤n:fk(x) – ≥εfor everyx∈A=nlim→∞
n
nα=
and
lim
n→∞
nαk≤n:fk(x) – ≥εfor everyx∈A=nlim→∞
n
forα> , so that{fk}statistically converges of orderαboth to and ,i.e.,Sα–limfk(x) =
andSα–limf
k(x) = , which is impossible.
Theorem . Let <α≤and{fk},{gk}be sequences of real-valued functions defined in a set A.
(i) IfSα–limf
k(x) =f(x)andc∈R,thenSα–limcfk(x) =cf(x).
(ii) IfSα–limf
k(x) =f(x)andSα–limgk(x) =g(x),then Sα–lim(f
k(x) +gk(x)) =f(x) +g(x).
Proof (i) The proof is clear in casec= . Suppose thatc= andSα–limf
k(x) =f(x), then
there existsε> such that
fk(x) –f(x)<
ε
|c| a.a.k(α),
and hence
cfk(x) –cf(x)<ε a.a.k(α).
This implies thatSα–limcf
k(x) =cf(x).
The proof of (ii) follows from the following inequalities:
nαk≤n:fk(x) +gk(x) –
f(x) +g(x)≥εfor everyx∈A
≤ nα
k≤n:fk(x) –f(x)≥
ε
for everyx∈A
+
nα
k≤n:gk(x) –g(x)≥
ε
for everyx∈A
.
It is easy to see that every convergent sequence of functions is statistically convergent of orderα, that is,c(f)⊂Sα(f) for each <α≤. But the converse of this does not hold. For example, the sequence{fk}defined by
fk(x) = ⎧ ⎨ ⎩
, k=n,
kx
+kx, k=n
is statistically convergent of orderαwithSα–limf
k(x) = forα>, but it is not
conver-gent.
Definition . Letαbe any real number such that <α≤ and let{fk}be a sequence of
functions on a setA. The sequence{fk}is a statistically Cauchy sequence of orderα(or
α-statistically Cauchy sequence) provided that for everyε> , there exists a numberN
(=N(ε,x)) such that
fk(x) –fN(x)<ε a.a.k(α),
i.e.,
lim
n
Theorem . Let{fk}be a sequence of functions defined on a set A.The following state-ments are equivalent:
(i) {fk}is a pointwiseα-statistically convergent sequence onA;
(ii) {fk}is aα-statistically Cauchy sequence onA;
(iii) {fk}is a sequence of functions for which there is a pointwise convergent sequence of
orderα,a sequence of functions{gk}such thatfk(x) =gk(x)a.a.k(α)for everyx∈A.
Proof (i)⇒(ii) Suppose thatSα–limf
k(x) =f(x) onAand letε> . Then|fk(x) –f(x)|<ε a.a.k(α) and ifNis chosen so that|fN(x) –f(x)|<ε, then we have
fk(x) –fN(x)≤fk(x) –f(x)+fN(x) –f(x)<
ε
+
ε
a.a.k(α)
for everyx∈A. Hence{fk}is anα-statistically Cauchy sequence.
Next, assume (ii) is true and chooseNso that the bandI= [fN(x) – ,fN(x) + ] contains fk(x)a.a.k(α) for everyx∈A. Also, apply (ii) to chooseMso thatI= [fM(x) –,fM(x) +]
containsfk(x)a.a.k(α) for everyx∈A. We assert that
I=I∩Icontainsfk(x)a.a.k(α) for everyx∈A;
for
k≤n:fk(x) /∈I∩Ifor everyx∈A
=k≤n:fk(x) /∈Ifor everyx∈A
∪k≤n:fk(x) /∈Ifor everyx∈A
so
lim
n→∞
nαk≤n:fk(x) /∈I∩I
for everyx∈A
≤ lim
n→∞
nαk≤n:fk(x) /∈Ifor everyx∈A
+ lim
n→∞
nαk≤n:fk(x) /∈I
for everyx∈A= .
Therefore,Iis a closed band of height less than or equal to that containsfk(x)a.a.k(α)
for everyx∈A. Now we proceed by choosingN() so thatI= [fN()(x) –,fN()(x) +]
containsfk(x)a.a.k(α), and by the preceding argument,I=I∩Icontainsfk(x)a.a.k
(α) for everyx∈AandIhas height less than or equal to . Continuing inductively, we construct a sequence{Im}∞m=of closed band such that for eachm,Im⊇Im+, the height ofIm is not greater than –mandfk(x)∈Im a.a.k(α) for everyx∈A. Thus there exists
a functionf(x), defined onA, such that{f(x)} is equal to∞m=Im. Using the fact that fk(x)∈Im a.a.k (α) for everyx∈A, we choose an increasing positive integer sequence {Tm}∞m=such that
nαk≤n:fk(x) /∈Imfor everyx∈A<
m ifn>Tm. ()
(gk(x)) by
gk(x) = ⎧ ⎨ ⎩
f(x) iffk(x) is a term ofzk(x),
fk(x) otherwise
for everyx∈A. Thenlimk→∞gk(x) =f(x) onA; for ifε>m > andk>Tm, then either fk(x) is a term of (zk(x)) orgk(x) =fk(x)∈ImonAand|gk(x) –fk(x)| ≤height ofIm≤–m
for everyx∈A. We also assert thatgk(x) =fk(x)a.a.k(α) for everyx∈A. To verify this, we
observe that ifTm<n≤Tm+, then
k≤n:fk(x)=gk(x) for everyx∈A
⊆k≤n:fk(x) /∈Imfor everyx∈A
.
So, by ()
nαk≤n:fk(x)=gk(x) for everyx∈A
≤
nαk≤n:fk(x) /∈Imfor everyx∈A<
m.
Hence, the limit is asn→ ∞andfk(x) =gk(x)a.a.k(α) for everyx∈A. Therefore,
(ii) implies (iii).
Finally, assume that (iii) holds, say fk(x) = gk(x) a.a.k (α) for every x ∈ A and
limk→∞gk(x) =f(x) onA. Letε> . Then for eachn,
k≤n:fk(x) –f(x)≥εfor everyx∈A
⊆k≤n:fk(x)=gk(x) for everyx∈A
∪k≤n:gk(x) –f(x)≥εfor everyx∈A
sincelimk→∞gk(x) =f(x) onA, the latter set contains a fixed number of integers, sayl= l(ε,x). Therefore,
lim
n→∞
nαk≤n:fk(x) –f(x)≥εfor everyx∈A
≤ lim
n→∞
nαk≤n:fk(x)= gk(x) for everyx∈A+nlim→∞
l nα =
becausefk(x) =gk(x)a.a.k(α) for everyx∈A. Hence|fk(x) –f(x)|<εa.a.k(α) for every
x∈A, so (i) holds and the proof is complete.
Corollary . If{fk}is a sequence of functions such that Sα–limfk(x) =f(x)on A,then{fk} has a subsequence{fk(n)(x)}such thatlimn→∞fk(n)(x) =f(x)on A.
Theorem . Let <α≤β≤.Then Sα(f)⊆Sβ(f)and the inclusion is strict for someα
Proof If <α≤β≤, then
nβk≤n:fk(x) –f(x)≥εfor everyx∈A
≤
nαk≤n:fk(x) –f(x)≥εfor everyx∈A
for everyε> and this gives thatSα(f)⊆Sβ(f). To show that the inclusion is strict, con-sider the sequence{fk}defined by
fk(x) = ⎧ ⎨ ⎩
, k=n,
kx
+kx, k=n,
n= , , , . . . ,x∈[, ].
Hence we can write for <α≤
nαk≤n:fk(x) – ≥εfor everyx∈[, ]
=
nαk≤n:fk(x)≥εfor everyx∈[, ]≤
√ n nα →.
ThenSβ–limf
k(x) = ,i.e.,x∈Sβ(f) for<β≤, butx∈/Sα(f) for <α≤.
If we takeβ= in Theorem ., then we obtain the following result.
Corollary . If a sequence of functions{fk}is statistically convergent of orderα,to the function f for some <α≤,then it is statistically convergent to the function f.
Definition . Letαbe any real number such that <α≤ and letpbe a positive real
number. A sequence of functions{fk}is said to be stronglyp-Cesàro summable of orderα
if there is a functionf such that
lim
n→∞
nα
n
k=
fk(x) –f(x)p= .
In this case, we writewα
p–limfk(x) =f(x) onA. The strongp-Cesàro summability of order
αreduces to the strongp-Cesàro summability forα= . The set of all stronglyp-Cesàro
summable sequences of functions of orderαwill be denoted bywα
p(f). We writewαo,p(f) in
casef(x) = .
Theorem . Let <α≤β≤and p be a positive real number.Then wα
p(f)⊆w
β
p(f)and the inclusion is strict for someαandβsuch thatα<β.
Proof Let the sequence{fk}be stronglyp-Cesàro summable of orderα. Then, givenαand
βsuch that <α≤β≤ and a positive real numberp, we may write
nβ
n
k=
fk(x) –f(x)p≤
nα
n
k=
fk(x) –f(x)p,
and this gives thatwα
p(f)⊆w
β
To show that the inclusion is strict, consider the sequence{fk}defined by
fk(x) = ⎧ ⎨ ⎩
+kx, k=n,
, k=n, x∈
,
k
.
Then
nβ
n
k=
fk(x) – p
≤ √
n nβ =
nβ–
since /(nβ–)→ asn→ ∞, thenwpβ–limfk(x) = ,i.e., the sequence{fk}is strongly
p-Cesàro summable of orderαfor
<β≤, but since
√ n
nα≤
nα
n
k=
fk(x) – p
and√n/nα→ ∞,n→ ∞, the sequence{f
k}is not stronglyp-Cesàro summable of order
αfor <α<
.
Corollary . Let <α≤β≤and p be a positive real number.Then
(i) ifα=β,thenwα
p(f) =w
β
p(f);
(ii) wα
p(f)⊆wp(f)for eachα∈(, ]and <p<∞.
Theorem . Let <α≤and <p<q<∞.Then wα
q(f)⊆wαp(f).
Proof Omitted.
Theorem . Letαandβbe fixed real numbers such that <α≤β≤and <p<∞.
If a sequence of functions{fk}is strongly p-Cesàro summable of orderαto the function f, then it is statistically convergent of orderβto the function f.
Proof For any sequence of functions{fk}defined onA, we can write
n
k=
fk(x) –f(x) p
≥k≤n:fk(x) –f(x)≥εfor everyx∈A·εp
and so that
nα
n
k=
fk(x) –f(x)p≥
nαk≤n:fk(x) –f(x)≥εfor everyx∈A·ε
p
≥
nβk≤n:fk(x) –f(x)≥εfor everyx∈A·ε
p.
Definition . Letαbe any real number such that <α≤. A sequence of functions
{fk}is said to be uniformly statistically convergent of orderαor uniformly (α-statistically
convergent sequence) to the functionf on a setAif, for everyε> ,
lim
n→∞
nαk≤n:fk(x) –f(x)≥εfor allx∈A= ,
i.e., for allx∈A,
fk(x) –f(x)<ε a.a.k(α). ()
In this case, we write
Sα–limfk(x) =f(x) uniformly onAorSαu–limfk(x) =f(x) onA.
The set of all uniformlyα-statistically convergent sequences will be denoted bySα
u(f).
Theorem . Let f and fk,for all k∈N,be continuous functions on A= [a,b]⊂R and
<α≤.Then Sα–limf
k(x) =f(x)uniformly on A if and only if Sα–limck= ,where ck=maxx∈A|fk(x) –f(x)|.
Proof Suppose thatSα–limf
k(x) =f(x) uniformly onA. Since|fk(x) –f(x)|is continuous
onAfor eachk∈N, it has absolute maximum value at some pointxk∈A,i.e., there exist x,x, . . .∈Asuch thatc=|f(x) –f(x)|,c=|f(x) –f(x)|, . . . ,etc.Thus we may write
ck=|fk(xk) –f(xk)|,k= , , . . . . From the definition of uniformα-statistical convergence,
we may write, for everyε> ,
fk(xk) –f(xk)<ε a.a.k(α).
Hence,Sα–limc
k= .
The necessity is trivial.
It follows from () that iflimfk(x) =f(x) uniformly onA, thenSα–limfk(x) =f(x)
uni-formly onA. But the converse is not true, for this consider the sequence defined by
fk(x) = ⎧ ⎨ ⎩
, k=n,
k
k+kx otherwise,
k= , , , . . . ,x∈[, ].
Then ifx∈[, ] andα∈[
, ], then{fk}is uniformlyα-statistically convergent tof(x) = on [, ] sinceSα–limc
k= , where
ck= max x∈[,]
fk(x) – = ⎧ ⎨ ⎩
, k=n,
k otherwise,
Corollary .
(i) limfk(x) =f(x)uniformly onA⇒limfk(x) =f(x)onA⇒Sα–limfk(x) =f(x)
pointwise onA. (ii) Sα–limf
k(x) =f(x)uniformly onA⇒Sα–limfk(x) =f(x)pointwise onA.
(iii) If <α≤β≤,thenSα
u(f)⊆S
β
u(f).
Definition . Letαbe any real number such that <α≤ and let{fk}be a sequence
of functions on a setA. The sequence{fk}is a uniformly statistically Cauchy sequence of
orderα(or uniformlyα-statistically Cauchy sequence) provided that for everyε> , there exists a numberN(=N(ε)) such that
fk(x) –fN(x)<ε a.a.k(α) for allx∈A,
i.e.,
lim
n→∞
nαk≤n:fk(x) –fN(x)≥εfor allx∈A= .
The proofs of the following two theorems are similar to those of Theorem . and The-orem ., therefore we give them without proof.
Theorem . Let <α≤and{fk},{gk}be sequences of real-valued functions defined on a set A.
(i) IfSα
u–limfk(x) =f(x)andc∈R,thenSαu–limcfk(x) =cf(x).
(ii) IfSα
u–limfk(x) =f(x)andSuα–limgk(x) =g(x),then Sα
u–lim(fk(x) +gk(x)) =f(x) +g(x).
Theorem . Letαbe any real number such that <α≤and let{fk}be a sequence of
functions on a set A.The following statements are equivalent: (i) {fk}is a uniformlyα-statistically convergent sequence onA;
(ii) {fk}is a uniformlyα-statistically Cauchy sequence onA;
(iii) {fk}is a sequence of functions for which there is a uniformly convergent sequence of
orderα,a sequence of functions{gk}such thatfk(x) =gk(x)a.a.k(α)for allx∈A.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
MC, MK, and ME have contributed to all parts of the article. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Mu¸s Alparslan University, Mu¸s, Turkey.2Department of Mathematics, Bitlis Eren University,
Bitlis, Turkey.3Department of Mathematics, Firat University, Elazıg, 23119, Turkey.
Received: 14 September 2012 Accepted: 30 January 2013 Published: 18 February 2013 References
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doi:10.1186/1687-1812-2013-33
