R E S E A R C H
Open Access
Invariant mean and a Korovkin-type
approximation theorem
Saleh Abdullah Al-Mezel
**Correspondence:
salmezel@kau.edu.sa
Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia
Abstract
In this paper we apply this form of convergence to prove some Korovkin-type approximation theorem by using the test functions 1,e–x,e–2x, which generalizes the results of Boyanov and Veselinov (Bull. Math. Soc. Sci. Math. Roum. 14(62):9-13, 1970).
MSC: 41A65; 46A03; 47H10; 54H25
Keywords: invariant mean;
σ
-convergence; Korovkin-type approximation theorem1 Introduction and preliminaries
Letcand∞denote the spaces of all convergent and bounded sequences, respectively, and
note thatc⊂∞. In the theory of sequence spaces, an application of the well-known Hahn-Banach extension theorem gave rise to the concept of the Hahn-Banach limit. That is, thelim
functional defined onccan be extended to the whole of∞and this extended functional
is known as the Banach limit. In , Lorentz [] used this notion of a generalized limit to define a new type of convergence, known as almost convergence. Later on, Raimi []
gave a slight generalization of almost convergence and named itσ-convergence. Before
proceeding further, we recall some notations and basic definitions used in this paper. Letσ be a mapping of the set of positive integers into itself. A continuous linear func-tionalϕdefined on the space∞of all bounded sequences is called an invariant mean (or aσ-mean;cf.[]) if it is non-negative, normal andϕ(x) =ϕ((xσ(n))).
A sequencex= (xk) is said to beσ-convergentto the numberLif and only if all of itsσ
-means coincide withL,i.e.,ϕ(x) =Lfor allϕ. A bounded sequencex= (xk) isσ-convergent
(cf.[]) to the numberLif and only iflimp→∞tpm=Luniformly inm, where
tpm=
xm+xσ(m)+xσ(m)+· · ·+xσp(m)
p+ .
We denote the set of allσ-convergent sequences byVσ and in this case we writexk→
L(Vσ) andLis called theσ-limit ofx. Note that aσ-mean extends the limit functional on
cin the sense thatϕ(x) =limxfor allx∈cif and only ifσ has no finite orbits (cf.[]) and
c⊂Vσ⊂∞.
If σ is a translation then the σ-mean is called a Banach limitandσ-convergence is reduced to the concept of almost convergence introduced by Lorentz [].
In [], the idea of statisticalσ-convergence is defined which is further applied to prove some approximation theorems in [] and [].
Ifm= , then we get (C, ) convergence, and in this case we writexk→(C, ), where = (C, )-limx.
Remark . Note that
(a) a convergent sequence is alsoσ-convergent; (b) aσ-convergent sequence implies(C, )convergence.
Example . Letσ(n) =n+ . Define the sequencez= (zn) by
zn=
ifnis odd, ifnis even.
Thenxisσ-convergent to / but not convergent.
LetC[a,b] be the space of all functionsf continuous on [a,b]. We know thatC[a,b] is a Banach space with the norm f∞:=supa≤x≤b|f(x)|, f ∈C[a,b]. Suppose thatTn:
C[a,b]→C[a,b]. We writeTn(f,x) forTn(f(t),x) and we say thatTis a positive operator
ifT(f,x)≥ for allf(x)≥.
The classical Korovkin approximation theorem states the following []: Let (Tn) be a
se-quence of positive linear operators fromC[a,b] intoC[a,b]. ThenlimnTn(f,x) –f(x)∞=
, for allf ∈C[a,b] if and only iflimnTn(fi,x) –fi(x)∞= , fori= , , , wheref(x) = ,
f(x) =xandf(x) =x.
Quite recently, such type of approximation theorem has been studied in [, ] and [] by usingλ-statistical convergence, while in [] lacunary statistical convergence has been
used. Boyanov and Veselinov [] have proved the Korovkin theorem onC[,∞) by
us-ing the test functions ,e–x,e–x. In this paper, we generalize the result of Boyanov and
Veselinov by using the notion ofσ-convergence. Our results also generalize the results of Mohiuddine [], in which the author has used almost convergence and the test functions ,x,x.
2 Korovkin-type approximation theorem
We prove the followingσ-version of the classical Korovkin approximation theorem.
Theorem . Let(Tk)be a sequence of positive linear operators from C(I)into C(I).Then,
for all f ∈C(I),
σ-lim
k→∞Tk(f;x) –f(x)∞= (.)
if and only if
σ-lim
k→∞Tk(;x) – ∞= , (.)
σ-lim
k→∞Tk
e–s;x–e–x∞= , (.)
σ-lim
k→∞
Tk
Proof Since each ,e–x, e–x belongs to C(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 (.), (.), 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+
M
δ (ψ).
Now, we operateTσk(n)(,x) for allnto this inequality sinceTσk(n)(f,x) is monotone and linear. We obtain
Tσk(n)(;x)
–ε–M
δ (ψ)
<Tσk(n)(;x)
f(s) –f(x)
<Tσk(n)(;x)
ε+M
δ (ψ)
.
Note thatxis fixed and sof(x) is a constant number. Therefore
–εTσk(n)(;x) –
M
δ Tσk(n)(ψ;x) <Tσk(n)(f;x) –f(x)Tσk(n)(;x) <εTσk(n)(;x) +
M
δ Tσk(n)(ψ;x). (.)
But
Tσk(n)(f;x) –f(x)
=Tσk(n)(f;x) –f(x)Tσk(n)(;x) +f(x)Tσk(n)(;x) –f(x)
= Tσk(n)(f;x) –f(x)Tσk(n)(;x)
+f(x)Tσk(n)(;x) –
. (.)
Using (.) and (.), we have
Tσk(n)(f;x) –f(x) <εTσk(n)(;x) +
M
δ Tσk(n)(ψ;x)
+f(x)Tσk(n)(;x) –
Now
Tσk(n)(ψ;x) =Tσk(n)
e–s–e–x;x=Tσk(n)
e–s– e–se–x+e–x;x
=Tσk(n)
e–s;x– e–xTσk(n)
e–s;x+e–xTσk(n)(;x) = Tσk(n)
e–s;x–e–x– e–x Tσk(n)
e–s;x–e–x
+e–x Tσk(n)(;x) –
.
Using (.), we obtain
Tσk(n)(f;x) –f(x) <εTσk(n)(;x) +
M
δ Tσk(n)
e–s;x–e–x
– e–x Tσk(n)
e–s;x–e–x+e–x Tσk(n)(;x) –
+f(x)Tσk(n)(;x) –
=ε[Tσk(n)(;x) – ] +ε+
M
δ Tσk(n)
e–s;x–e–x
– e–x Tσk(n)
e–s;x–e–x+e–x Tσk(n)(;x) –
+f(x)Tσk(n)(;x) –
.
Sinceεis arbitrary, we can write
Tσk(n)(f;x) –f(x)≤ε Tσk(n)(;x) –
+M
δ Tσk(n)
e–s;x–e–x
– e–x Tσk(n)
e–s;x–e–x+e–x Tσk(n)(;x) –
+f(x) Tσk(n)(;x) –
.
Therefore
Tσk(n)(f;x) –f(x)
≤ε+ (ε+M)Tσk(n)(;x) – +
M
δ e –xT
σk(n)(;x,y) –
+M
δ Tσk(n)
e–s;x–e–x+M
δ e –xT
σk(n)
e–s;x–e–x
≤ε+
ε+M+M
δ
Tσk(n)(;x) – +
M
δ e –xT
σk(n)(;x) –
+M
δ Tσk(n)
e–s;x–e–x+M
δ Tσk(n)
e–s;x–e–x
since|e–x| ≤ for allx∈I. Now, takingsup x∈I
Tσk(n)(f;x) –f(x)∞≤ε+KTσk(n)(;x) – ∞+Tσk(n)
e–s;x–e–x∞
+Tσk(n)
whereK=max{ε+M+δM ,δM}. Now writing
Dn,p(f,x) =
p
p–
k=
Tσk(n)(f,x),
we get
Dn,p(f,x) –f(x)∞≤
+Mb
δ +M
Dn,p(,x) – ∞
+Mb
δ Dn,p(t,x) –e –x
∞+
M
δ Dn,p
t,x–e–x∞.
Lettingp→ ∞and using (.), (.), (.), we get
lim
p→∞Dn,p(f,x) –f(x)∞= , uniformly inn.
In the following example we construct a sequence of positive linear operators satisfying the conditions of Theorem . but not satisfying the conditions of the Korovkin theorem of Boyanov and Veselinov [].
Example . Consider the sequence of classical Baskakov operators []
Vn(f;x) := ∞
k=
f
k n
n– +k k
xk( +x)–n–k,
where ≤x,y<∞.
Let the sequence (Ln) be defined byLn:C(I)→C(I) withLn(f;x) = ( +zn)Vn(f;x), where
znis defined as above. Since
Ln(;x) = ,
Ln
e–s;x= +x–xe–n–n,
Ln
e–s;x= +x–xe–n–n,
and the sequence (Pn) satisfies the conditions (.), (.) and (.). Hence we have
σ-limLn(f,x) –f(x)∞= .
On the other hand, we getLn(f, ) = ( +zn)f() sinceLn(f, ) =f(), and hence
Ln(f,x) –f(x)∞≥Ln(f, ) –f()=znf().
We see that (Ln) does not satisfy the classical Korovkin theorem sincelim supn→∞zn
3 A consequence
Now we present a slight general result.
Theorem . Let(Tn)be a sequence of positive linear operators on C(I)such that
lim
n supm
n
n–
k=
Tn–Tσk(m)= .
If
σ-lim
n Tn
e–νs,x–e–νx∞= (ν= , , ), (.)
then,for any function f ∈C(I)bounded on the real line,we have
lim
n Tn(f,x) –f(x)∞= . (.)
Proof From Theorem ., we have that if (.) holds, then
σ-lim
n Tn(f,x) –f(x)∞= ,
which is equivalent to
lim
n
sup
m
Dm,n(f,x) –f(x) ∞= .
Now
Tn–Dm,n=Tn–
n
n–
k=
Tσk(m)
=
n
n–
k=
(Tn–Tσk(m)).
Therefore
Tn–sup m
Dm,n=sup m
n
n–
k=
(Tn–Tσk(m)).
Hence, using the hypothesis, we get
lim
n Tn(f,x) –f(x)∞=limn
sup
m
Dm,n(f,x) –f(x) ∞= ,
that is, (.) holds.
Competing interests
The author declares that they have no competing interests.
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doi:10.1186/1029-242X-2013-103
