R E S E A R C H
Open Access
Statistical summability
(
C
, )
and a Korovkin
type approximation theorem
Syed Abdul Mohiuddine
1, Abdullah Alotaibi
1*and Mohammad Mursaleen
2*Correspondence: [email protected] 1Department 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
The concept of statistical summability (C, 1) has recently been introduced by Móricz [Jour. Math. Anal. Appl.275, 277-287 (2002)]. In this paper, we use this notion of summability to prove the Korovkin type approximation theorem by using the test functions 1,e–x,e–2x. We also give here the rate of statistical summability (C, 1) and apply the classical Baskakov operator to construct an example in support of our main result.
MSC: 41A10; 41A25; 41A36; 40A30; 40G15
Keywords: statistical convergence; statistical summability (C, 1); positive linear operator; Korovkin type approximation theorem
1 Introduction and preliminaries
In , Fast [] presented the following definition of statistical convergence for sequences of real numbers. LetK⊆N, the set of all natural numbers andKn={k≤n:k∈K}. Then
thenatural densityofK is defined byδ(K) =limnn–|Kn| if the limit exists, where the
vertical bars indicate the number of elements in the enclosed set. The sequencex= (xk) is
said to bestatistically convergenttoLif for every> , the setK:={k∈N:|xk–L| ≥}
has natural density zero,i.e., for each> ,
lim
n
n{j≤n:|xj–L| ≥}= .
In this case, we writeL=st-limx. Note that every convergent sequence is statistically con-vergent but not conversely. Define the sequencew= (wn) by
wn=
ifn=k,k∈N,
otherwise. (.)
Thenxis statistically convergent to but not convergent.
Recently, Móricz [] has defined the concept of statistical summability (C, ) as follows: For a sequencex= (xk), let us writetn=n+
n
k=xk. We say that a sequencex= (xk) is
In the following example, we exhibit that a sequence is statistically summable (C, ) but not statistically convergent. Define the sequencex= (xk) by
xk=
⎧ ⎪ ⎨ ⎪ ⎩
ifk=m–m,m–m+ , . . . ,m– , –m ifk=m,m= , , , . . . ,
otherwise.
(.)
Then
tn=
n+
n
k= xk
=
s+
n+ ifn=m–m+s;s= , , , . . . ,m– ;m= , , . . . ,
otherwise. (.)
It is easy to see that limn→∞tn = and hence st-limn→∞tn= , i.e., a sequence x=
(xk) is statistically summable (C, ) to . On the other hand st-lim infk→∞xk = and
st-lim supk→∞xk= , since the sequence (m)∞m= is statistically convergent to . Hence x= (xk) is not statistically convergent.
LetC[a,b] be the space of all functionsf continuous on [a,b]. We know thatC[a,b] is a Banach space with norm
f∞:= sup
x∈[a,b]
f(x), f∈C[a,b].
The classical Korovkin approximation theorem is stated as follows []:
Let (Tn) be a sequence of positive linear operators from C[a,b] into C[a,b]. Then
limnTn(f,x) –f(x)∞ = , for all f ∈C[a,b] if and only iflimnTn(fi,x) –fi(x)∞= ,
fori= , , , wheref(x) = ,f(x) =xandf(x) =x.
Recently, Mohiuddine [] has obtained an application of almost convergence for single sequences in Korovkin-type approximation theorem and proved some related results. For the function of two variables, such type of approximation theorems are proved in [] by us-ing almost convergence of double sequences. Quite recently, in [] and [] the Korovkin type theorem is proved for statisticalλ-convergence and statistical lacunary summability, respectively. For some recent work on this topic, we refer to [, , , , , ]. Boyanov and Veselinov [] have proved the Korovkin theorem onC[,∞) by using the test functions ,e–x,e–x. In this paper, we generalize the result of Boyanov and Veselinov by using the
notion of statistical summability (C, ) and the same test functions ,e–x,e–x. We also give
an example to justify that our result is stronger than that of Boyanov and Veselinov [].
2 Main result
LetC(I) be the Banach space with the uniform norm · ∞of all real-valued two
dimen-sional continuous functions onI= [,∞); provided thatlimx→∞f(x) is finite. Suppose that
Ln:C(I)→C(I). We writeLn(f;x) forLn(f(s);x); and we say thatLis a positive operator if
L(f;x)≥ for allf(x)≥.
Theorem A Let(Tk)be a sequence of positive linear operators from C(I)into C(I). Then
for all f ∈C(I)
st-lim
k→∞Tk(f;x) –f(x)∞=
if and only if
st-lim
k→∞Tk(;x) – ∞= ,
st-lim
k→∞
Tk
e–s;x–e–x∞= ,
st-lim
k→∞
Tk
e–s;x–e–x∞= .
Now we prove the following result by using the notion of statistical summability (C, ). Theorem . Let(Tk)be a sequence of positive linear operators from C(I)into C(I). Then
for all f ∈C(I)
C(st)-lim
k→∞Tk(f;x) –f(x)∞= (.)
if and only if
C(st)-lim
k→∞Tk(;x) – ∞= , (.)
C(st)-lim
k→∞Tk
e–s;x–e–x∞= , (.)
C(st)-lim
k→∞
Tk
e–s;x–e–x∞= . (.)
Proof Since each of ,e–x,e–xbelongs toC(I), conditions (.)-(.) follow immediately
from (.). Letf ∈C(I). Then there exists a constantM> such that|f(x)| ≤Mforx∈I. Therefore,
f(s) –f(x)≤M, –∞<s,x<∞. (.)
It is easy to prove that for a givenε> there is aδ> such that
f(s) –f(x)<ε, (.)
whenever|e–s–e–x|<δfor allx∈I.
Using (.), (.), and puttingψ=ψ(s,x) = (e–s–e–x), we get f(s) –f(x)<ε+M
δ (ψ), ∀|s–x|<δ.
This is,
–ε–M
δ (ψ) <f(s) –f(x) <ε+
Now, applying the operatorTk(;x) to this inequality, sinceTk(f;x) is monotone and linear,
we obtain
Tk(;x)
–ε–M
δ (ψ)
<Tk(;x)
f(s) –f(x)<Tk(;x)
ε+M
δ (ψ)
.
Note thatxis fixed and sof(x) is a constant number. Therefore,
–εTk(;x) –
M
δ Tj,k(ψ;x) <Tk(f;x) –f(x)Tk(;x) <εTk(;x) +
M
δ Tk(ψ;x). (.)
Also,
Tk(f;x) –f(x) =Tk(f;x) –f(x)Tk(;x) +f(x)Tk(;x) –f(x)
=Tk(f;x) –f(x)Tk(;x) +f(x)
Tk(;x) –
. (.)
It follows from (.) and (.) that
Tk(f;x) –f(x) <εTk(;x) +
M
δ Tk(ψ;x) +f(x)
Tk(;x) –
. (.)
Now
Tk(ψ;x) =Tk
e–s–e–x;x
=Tk
e–s– e–se–x+e–x;x
=Tk
e–s;x– e–xTk
e–s;x+e–xTk(;x)
=Tk
e–s;x–e–x– e–xTk
e–s;x–e–x+e–xTk(;x) –
.
Using (.), we obtain
Tk(f;x) –f(x) <εTk(;x) +
M δ
Tk
e–s;x–e–x– e–xTk
e–s;x–e–x
+e–xTk(;x) –
+f(x)Tk(;x) –
=εTk(;x) –
+ε+M
δ
Tk
e–s;x–e–x– e–xTk
e–s;x–e–x
+e–xTk(;x) –
+f(x)Tk(;x) –
.
Therefore,
Tk(f;x) –f(x)≤ε+ (ε+M)Tk(;x) – +
M δ Tk
e–s;x–e–x
+M
δ e –xT
k
e–s;x–e–x+M
δ e –xT
k(;x) –
≤ε+
ε+M+M
δ
Tk(;x) – +
M δ Tk
e–s;x–e–x
+M
δ Tk
since|e–x| ≤ for allx∈I. Now takingsup
x∈I, we get
Tk(f;x) –f(x)∞≤ε+KTk(;x) – ∞+Tk
e–s;x–e–x
∞
+Tk
e–s;x–e–x∞, (.) whereK=max{ε+M+δM ,δM ,δM }.
Now replacingTk(·,x) by m+
m
k=Tk(·,x) and then byBm(·,x) in (.) on both sides.
For a givenr> chooseε> such thatε<r. Define the following sets
D=m≤n:Bm(f,x) –f(x)∞≥r
,
D=
m≤n:Bm(,x) – ∞≥
r–ε
K
,
D=
m≤n:Bm(t,x) –e–x∞≥
r–ε
K
,
D=
m≤n:Bm
t,x–e–x∞≥r–ε
K
.
ThenD⊂D∪D∪D, and soδ(D)≤δ(D) +δ(D) +δ(D). Therefore, using conditions
(.)-(.), we get
C(st)-lim
n Tn(f,x) –f(x)∞= .
This completes the proof of the theorem.
3 Rate of statistical summability(C, 1)
In this section, we study the rate of weighted statistical convergence of a sequence of pos-itive linear operators defined fromC(I) intoC(I).
Definition . Let (an) be a positive nonincreasing sequence. We say that the sequence
x= (xk) is statistically summable (C, ) to the numberLwith the rateo(an) if for every
ε> ,
lim
n
an
m≤n:|tm–L| ≥ε= .
In this case, we writexk–L=C(st) –o(an).
Now, we recall the notion of modulus of continuity. The modulus of continuity off ∈ C(I), denoted byω(f,δ) is defined by
ω(f,δ) = sup |x–y|<δ
f(x) –f(y).
It is well known that
f(x) –f(y)≤ω(f,δ)
|e–y–e–x|
δ +
. (.)
Theorem . Let(Tk)be a sequence of positive linear operators from C(I)into C(I).
Sup-pose that
(i) Tk(;x) – ∞=C(st) –o(an),
(ii) ω(f,λk) =C(st) –o(bn), whereλk=
Tk(ϕx;x)andϕx(y) = (e–y–e–x).
Then for all f ∈C(I), we have
Tk(f;x) –f(x)∞=C(st) –o(cn),
where cn=max{an,bn}.
Proof Letf ∈C(I) andx∈I. From (.) and (.), we can write
Tk(f;x) –f(x)≤Tkf(y) –f(x);x
+f(x)Tk(;x) –
≤Tk
|e–y–e–x|
δ + ;x
ω(f,δ) +f(x)Tk(;x) –
≤Tk
+
δ
e–y–e–x;x
ω(f,δ) +f(x)Tk(;x) –
≤
Tk(;x) +
δTk(ϕx;x)
ω(f,δ) +f(x)Tk(;x) –
≤ω(f,δ)Tk(;x) – +|f|Tk(;x) – +ω(f,δ)
+
δω(f,δ)Tk(ϕx;x).
Putδ=λk=
Tk(ϕx;x). Hence, we get
Tk(f;x) –f(x)∞≤ fTk(;x) – ∞+ ω(f,λk) +ω(f,λk)Tk(;x) – ∞
≤KTk(;x) – ∞+ω(f,λk) +ω(f,λk)Tk(;x) – ∞
, whereK=max{f, }. Now replacingTk(·;x) byn+
n
k=Tk(·;x) =Ln(·;x)
Ln(f;x) –f(x)∞≤KLn(;x) – ∞+ω(f,λk) +ω(f,λk)Tk(;x) – ∞
.
Using the Definition ., and conditions (i) and (ii), we get the desired result. This completes the proof of the theorem.
4 Example and the concluding remark
In the following, we construct an example of a sequence of positive linear operators sat-isfying the conditions of Theorem . but does not satisfy the conditions of the Korovkin approximation theorem due to of Boyanov and Veselinov [] and the conditions of Theo-rem A.
Consider the sequence of classical Baskakov operators []
Vn(f;x) :=
∞
k= f
k n
n– +k
k
xk( +x)–n–k,
LetLn:C(I)→C(I) be defined by
Ln(f;x) = ( +xn)Vn(f;x),
where the sequencex= (xn) is defined by (.). Note that this sequence is statistically
summable (C, ) to but neither convergent nor statistically convergent. Now
Vn(;x) = ,
Vn
e–s;x= +x–xe–n–n,
Vn
e–s;x= +x–xe–n–n,
we have that the sequence (Ln) satisfies the conditions (.), (.), and (.). Hence, by
Theorem ., we have
C(st)-lim
n→∞Ln(f) –f∞= .
On the other hand, we getLn(f; ) = ( +xn)f(), sinceVn(f; ) =f(), and hence
Ln(f;x) –f(x)∞≥Ln(f; ) –f()=xnf().
We see that (Ln) does not satisfy the conditions of the theorem of Boyanov and Veselinov
as well as of Theorem A, since (xn) is neither convergent nor statistically convergent.
Hence, our Theorem . is stronger than that of Boyanov and Veselinov [] as well as Theorem A.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All the authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia. 2Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India.
Acknowledgements
The authors would like to thank the Deanship of Scientific Research at King Abdulaziz University, Saudi Arabia, for its financial support under Grant No. 409/130/1432.
Received: 27 April 2012 Accepted: 19 July 2012 Published: 6 August 2012 References
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doi:10.1186/1029-242X-2012-172
