R E S E A R C H
Open Access
Statistical weighted
A
-summability with
application to Korovkin’s type approximation
theorem
Syed Abdul Mohiuddine
**Correspondence:
[email protected] Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia
Abstract
We introduce the notion of statistical weightedA-summability of a sequence and establish its relation with weightedA-statistical convergence. We also define
weighted regular matrix and obtain necessary and sufficient conditions for the matrix Ato be weighted regular. As an application, we prove the Korovkin type
approximation theorem through statistical weightedA-summability and using the BBH operator to construct an illustrative example in support of our result.
MSC: 41A10; 41A25; 41A36; 40A30
Keywords: statistical convergence; statistical weightedA-summability; regular matrix; positive linear operator; Korovkin theorem
1 Introduction and preliminaries
The term ‘statistical convergence’ was first presented by Fast []; it is a generalization of the concept of ordinary convergence. Actually, a root of the notion of statistical convergence can be detected by Zygmund [] (also see []), where he used the term ‘almost conver-gence’ which turned out to be equivalent to the concept of statistical convergence. Sta-tistical convergence was further investigated by Schoenberg [], Šalát [], Fridy [], and Connor [].
Recall the definition of natural (or asymptotic) density as follows: Suppose thatE⊆N:=
{, , . . .}andEn={k≤n:k∈E}. Then
δ(E) =lim
n
n|En|
is called thenatural densityofE provided that the limit exists, where| · | denotes the cardinality of the enclosed set. A sequences= (sk) is said to bestatistically convergent,
shortlyS-convergent, [] toL, in symbols, we shall writeS-limx=L, ifδ(K) = for every
> , where
K:=
k∈N:|sk–L| ≥
,
equivalently,
lim
n n
–k≤n:|s
k–L| ≥= .
We remark that every convergent sequence is statistically convergent but its converse is not always valid.
In , the concept of weighted statistical convergence was defined and studied by Karakaya and Chishti [] and further modified by Mursaleenet al.[] in .
Letp= (pk) be a sequence of nonnegative numbers such thatp> and
Pn= n
k=
pk→ ∞ asn→ ∞.
The lower and upper weighted densities ofE⊆Nare defined by
δN¯(E) =lim infn
Pn
{k≤Pn:k∈E}
and
δN¯(E) =lim sup
n
Pn
{k≤Pn:k∈E},
respectively. We say thatEhasweighted density, denoted byδN¯(E), if the limits of both
above densities exist and are equal, that is, one writes
δN¯(E) =lim
n
Pn
{k≤Pn:k∈E}.
The sequences= (sk) is said to beweighted statistically convergent(orSN¯-convergent)
toLif for every> , the set{k∈N:pk|sk–L| ≥}has weighted density zero,i.e.
lim
n
Pn
k≤Pn:pk|sk–L| ≥= .
In this case we writeL=SN¯-lims.
Remark . Ifpk= for allkthen weighted statistical convergence is reduced to statistical
convergence.
In , Belen and Mohiuddine [] presented a generalization of this notion through de la Vallée-Poussin mean and called this weightedλ-statistical convergence (in short,SNλ¯
-convergence). Recently, the notion was modified by Ghosal [] by adding the condition
lim infpk> .
LetXandYbe two sequence spaces and letA= (an,k) be an infinite matrix. If for each
s= (sk) inXthe series
Ans=
k
an,ksk=
∞
converges for eachn∈Nand the sequenceAs= (Ans) belongs toY, then we say that matrix
AmapsX intoY. By the symbol (X,Y) we denote the set of all matrices which mapX
intoY.
A matrixA(or a matrix mapA) is called regular ifA∈(c,c), where the symbolcdenotes the spaces of all convergent sequences, and
lim
n Ans=limk sk
for alls∈c. The well-known Silverman-Toeplitz theorem (see []) asserts thatA= (an,k)
is regular if and only if
(i) supnk|an,k|<∞;
(ii) limnan,k= for eachk;
(iii) limn
kan,k= .
Kolk [] extended the definition of statistical convergence with the help of the non-negative regular matrixA= (an,k), which he called itA-statistical convergence. For any
nonnegative regular matrixA, we say that a sequences= (sk) isA-statistically convergent,
shortlySA-convergent, toLprovided that for every> we have
lim
n
k:|sk–L|≥
an,k= .
Edely and Mursaleen [] introduced statisticalA-summability (or,AS-summability) and
showed thatSA-convergence impliesAS-summability under the assumption of bounded
sequence but the converse is not valid always.
2 Statistical weightedA-summability
In the present section we introduce the notion of statistical weightedA-summability and prove that this method of summability is stronger than the weightedA-statistically con-vergent notion. We also define and characterize a weighted regular matrix.
Lets= (sk)k∈Nbe a sequence of real or complex numbers. The weighted meansσnhere
are of the form
σn=
Qn n
k=
qksk (n≥),
whereq= (qk)k∈Nis a given sequence of nonnegative numbers such thatlim infkqk> and
Qthe sequence withQn=
n
k=qk= for alln≥. If
lim
n
Qnn
k=
qksk–L
= ,
then we say thats= (sk)k∈Nisweighted summable, calledcN¯-summable, to some numberL.
In symbols, we shall writecN¯-lims=LandcN¯ denotes the space of all weighted summable sequences. In particular, if we takeqk= for allkthencN¯-summability is reduced to (C,
Liflimn→∞σn=L, where
σn=
n
n
k= sk.
Definition . A sequence s= (sk)k∈N is said to be weighted A-summable if the A
-transform ofsis weighted summable. It is said to beweighted A-summableto Lif the
A-transform ofsis weighted summable toL, that is,
lim m Qm m n= ∞ k=
qnan,ksk–L
= .
For convenience, we shall use the convention that
ANm¯(s) =
Qm m n= ∞ k=
qnan,ksk.
Definition . The matrixA(or a matrix mapA) is said to beweighted regular matrix, orweighted regular method, ifAs∈cN¯ for alls= (s
k)∈cwithcN¯-limAs=limsand one
denotes this byA∈(c,cN¯). Clearly,A∈(c,cN¯) means thatAN¯
m(s) exists for eachm∈Nand
eachs∈cand thatAN¯
m(s)→L(m→ ∞) wheneversk→L(k→ ∞).
We prove the following characterization of a weighted regular matrix.
Theorem . The matrix A= (an,k)is weighted regular,that is,A∈(c,cN¯),if and only if
sup m ∞ k= Qm m n= qnan,k
<∞; ()
lim m Qm m n=
qnan,k= for each k; ()
lim m Qm m n= ∞ k=
qnan,k= . ()
Proof Sufficiency. Let the conditions ()-() hold, and suppose thatsk∈cwithsk→Las
k→ ∞. Then for each> there existsN∈Nsuch that|sk|<|L|+fork>N. One writes
Qm m n= ∞ k=
qnan,ksk=
Qm ∞ k= m n=
qnan,ksk
= Qm m– k= m n=
qnan,ksk+
Qm
∞
k=m–
m
n=
qnan,ksk.
Therefore Qm m n= ∞ k=
qnan,ksk
≤ s
m– k= Qm m n=
qnan,k+
Takingm→ ∞and using () and (), we obtain
Qm m
n=
∞
k=
qnan,ksk
≤ |L|+
and consequently
lim
m
Qm m
n=
∞
k=
qnan,ksk=L=limsk (sincewas arbitrary).
This shows thatAis weighted regular.
Necessity. LetA∈(c,cN¯). Takingek,e∈c, whereekis the sequence with in placekand
elsewhere ande= (, , , . . . ), then theA-transforms of the sequenceekandebelong to
cN¯ and hencee
k∈cgives condition () ande∈cproves the validity of ().
Let us write
ANm¯(s) =
Qm m
n=
qnβn(s), βn(s) =
∞
k= an,ksk.
Clearly,βn∈c (the linear space of all continuous linear functionals ofc) and soANm¯ ∈c.
SinceAis almost regular,AN¯
m(x) exists for eachm∈Nand eachs∈candcN¯-limAm(s) =
L=limsk. It follows that (AmN¯(s)) is bounded fors∈c. Hence (A
¯
N
m) is bounded by the
uniform boundedness principle.
For eachb∈Z+, the positive integers, we definex= (x
k) by
xk=
sgnmn=qnan,k for ≤k≤b,
fork>b. Thenx∈c,x= , and also
ANm¯(x)=
Qm b
k=
m
n= qnan,k
.
Hence
ANm¯(x)≤AmN¯x=ANm¯.
Therefore
Qm
∞
k=
m
n= qnan,k
≤A
¯
N m,
so () is valid.
Definition . LetA= (an,k) be a nonnegative weighted regular matrix and letE⊆N.
Then theweighted A-densityofEis given by
δA¯
N(E) =limm
Qm m
n=
k∈E
qnan,k
provided that the limit exists. A sequencex= (xk) of real or complex numbers is said to be
weighted A-statistically convergent, denoted bySN¯
A-convergent, toLif for every>
δAN¯E()= , where
E() =k∈N:|xk–L| ≥
. In symbols, we shall writeSNA¯-limx=L.
Definition . LetA= (an,k) be a nonnegative weighted regular matrix. A sequences=
(sk) of real or complex numbers is said to bestatistically weighted A-summable, denoted
byAN¯
S-summable, toL, in symbols, we shall writeA
¯
N
S-lims=L, if the following equality
holds for each> :
δ(E) = ,
whereE={m∈N:|Tm–L| ≥}and
Tm=ANm¯(s) =
Qm m
j=
∞
k= qjaj,ksk,
equivalently, we can write
lim
n
nm≤n:|Tm–L| ≥= .
Thus, a sequences= (sk) isANS¯-summable toLif and only ifA
¯
N
m(s) isS-convergent toL.
Theorem . If a sequence s= (sk)is bounded and weighted A-statistically convergent to
L then it is weighted A-summable to L and hence statistically weighted A-summable to L but not conversely.
Proof Let (sk) be bounded and weightedA-statistically convergent toL, and letE() ={k∈
N:|sk–L| ≥}. Then
Qm m
n=
∞
k=
qnan,ksk–L
=
Qm m
n=
∞
k=
qnan,k(sk–L) +L
Qm m
n=
∞
k=
qnan,k–
≤ Qm m n= ∞ k=
qnan,k(sk–L)
+|L|
Qm m n= ∞ k=
qnan,k–
≤ Qm m n=
k∈E()
qnan,k(sk–L)
+ Qm m n=
k∈/E()
qnan,k(sk–L)
+|L|
Qm m n= ∞ k=
qnan,k–
≤sup
k
|sk–L|
Qm m n=
k∈E()
qnan,k+
Qm m n=
k∈/E() qnan,k
+|L|
Qm m n= ∞ k=
qnan,k–
.
Using the definition of weighted A-statistical convergence and the conditions of a weighted regularity of the matrixA, we have
lim m Qm m n= ∞ k=
qnan,ksk–L
= (sincewas arbitrary),
that is,sis weightedA-summable toL. Hence
S-lim
m Qm m n= ∞ k=
qnan,ksk–L
= .
This shows that the sequencesisANS¯-summable toL.
Example . Let us takeAas Cesàro matrix (or, (C, )-matrix) and is defined as follows:
an,k=
n if ≤k≤n,
ifk>n.
Consider a bounded sequences= (sk) which is defined by
sk=
ifkis odd, ifkis even,
and suppose also that qn= for alln= , . . . ,mand soQm =m. Then we see thatsis
weightedA-summable to / and hence statistically weightedA-summable to the same limit but not weightedA-statistically convergent.
3 Application
In this final section, we apply our previous notion of summability,i.e. ANS¯-summability, to obtain the Koronkin type approximation theorem.
,x,x(or other equivalent test functions). Many mathematicians extended the Korovkin’s
type approximation theorems by using various test functions in several setup, including Banach spaces, abstract Banach lattices, function spaces, Banach algebras, and so on. First of all, Gadjiev and Orhan [] established the classical Korovkin theorem through statis-tical convergence and displayed an interesting example in support of the result. Recently, Korovkin’s type theorems have been obtained by Mohiuddine [] and Edelyet al.[] for almost convergence andλ-statistical convergence, respectively. The authors of [] es-tablished these types of approximation theorem in weightedLpspaces, where ≤p<∞,
throughA-summability, which is stronger than ordinary convergence. For these type of approximation theorems and related concepts, one may refer to [–] and the refer-ences therein.
We use the notationCB(D) to denote the space of all continuous and bounded
real-valued functions onD=I×Iequipped with the following norm:
hCB(D):= sup (x,y)∈D
h(x,y), h∈CB(D),
whereI= [,∞). Supposehis a real-valued function onDsuch that
h(g,r) –h(x,y)≤ω∗
h;
g
+g – x
+x
+
r
+r– y
+y
,
and the space of such functions is denoted byHω∗(D). In this caseω∗is used for the
mod-ulus of continuity and is defined by
ω∗(h;δ) = sup (g,r),(x,y)∈K
h(g,r) –h(x,y):(g–x)+ (r–y)≤δ,δ> .
We remark that any functionh∈Hω∗(D) is continuous and bounded onD, and a necessary
and sufficient condition forh∈Hω∗(D) is that
lim
δ→ω
∗(h;δ) = .
We are writing the Korovkin type approximation theorem of Çakar and Gadjiev [] through the usual convergence for the following test functions:
h(g,r) = , h(g,r) = g
+g, h(g,r) = r
+r,
and
h(g,r) =
g
+g
+
r
+r
.
Theorem . Let(Jk)be a sequence of positive linear operators(PLO)from Hω∗(D)into
CB(D).Then
lim
k→∞Jk(h;x,y) –h(x,y)CB(D)=
if and only if
lim
k→∞Jk(hi;x,y) –hiCB(D)= , where i= , , , .
ForSA-convergence andAS-summability, proofs are in [] and [], respectively. We
prove the following Çakar and Gadjiev [] type theorem through a statistically weighted
A-summability method.
Theorem . Let A= (aj,k)be a nonnegative weighted regular matrix and(Jk)be a
se-quence of PLO from Hω∗(D)into CB(D).Then
S-lim
Qm m j= ∞ k=
qjaj,kJk(h;x,y) –h(x,y)
CB(D)
= ∀h∈Hω∗(D)
()
if and only if
S-lim
Qm m j= ∞ k=
qjaj,kJk(;x,y) –
CB(D)
= , ()
S-lim
Qm m j= ∞ k= qjaj,kJk
g
+g;x,y
– x +x
CB(D)
= , ()
S-lim
Qm m j= ∞ k= qjaj,kJk
r
+r;x,y
– y +y
CB(D)
= , ()
S-lim
Qm m j= ∞ k= qjaj,kJk
g
+g
+
r
+r
;x,y
–
x
+x
+
y
+y
CB(K)
= . ()
Proof The conditions ()-() follow immediately from () by taking into account that each of the functionshibelongs toHω∗(K), wherei= , , , . Leth∈Hω∗(K) and (x,y)∈Dbe
fixed. Letε> be given. Then there existδ,δ> such that
h(g,r) –h(x,y)<
holds for all (g,r) inDsatisfying the conditions:
+gg – x +x
<δ and
+rr– y +y
<δ.
ConsiderD(δ) of the form
D(δ) :=
(g,r)∈D:
g
+g – x
+x
+
r
+r– y
+y
<δ
whereδ=min{δ,δ}. Therefore, we obtain
h(g,r) –h(x,y)=h(g,r) – (x,y)χD(δ)(g,r) +h(g,r) –h(x,y)χD\D(δ)(g,r)
≤+ AχD\D(δ)(g,r), ()
whereχKstands for the characteristic function ofKandA=hCB(D). Also, we obtain
χD\D(δ)(g,r)≤
δ
g
+g– x
+x
+
δ
r
+r– y
+y
. ()
The inequalities () and () give
h(g,r) –h(x,y)≤+A
δ
g
+g – x
+x
+
r
+r– y
+y
. ()
By a direct computation, we see that
Jk(h;x,y) –h(x,y)≤+NJk(h;x,y) –h(x,y)
+Jk(h;x,y) –h(x,y)+Jk(h;x,y) –h(x,y)
+Jk(h;x,y) –h(x,y), ()
where
N:=ε+A+A
δ .
Consequently, by writing
Qm m
j=
∞
k=
qjaj,kJk(hi;x,y)
instead ofJk(hi;x,y) (i= , , , ) and considering the supremum over (x,y)∈D, one
ob-tains
Qm m
j=
∞
k=
qjaj,kJk(h;x,y) –h(x,y)
CB(D)
≤+N
Qm m
j=
∞
k=
qjaj,kJk(h;x,t) –h(x,y)
CB(D)
+
Qm m
j=
∞
k=
qjaj,kJk(h;x,y) –h(x,y)
CB(D)
+
Qm m
j=
∞
k=
qjaj,kJk(h;x,y) –h(x,y)
CB(D)
+
Qm m
j=
∞
k=
qjaj,kJk(h;x,y) –h(x,y)
CB(D)
For a givent> choose> such that<t, we shall define the following sets:
V= m∈N:
Qm m
j=
∞
k=
qjaj,kJk(h;x,y) –h(x,y)
CB(D) ≥t
,
and
Vi= m∈N:
Qm m
j=
∞
k=
qjaj,kJk(hi;x,y) –hi(x,y)
CB(D) ≥t–
N
,
wherei= , , , . This shows thatV⊂i=Viand so
δ(V)≤δ(V) +δ(V) +δ(V) +δ(V).
Hence, using assumptions ()-(), we get
S-lim
Qm m
j=
∞
k=
qjaj,kJk(h;x,y) –h(x,y)
CB(D)
= .
Finally, we conclude our work by the following illustration, Example ., which shows that Theorem . is stronger than Theorem ..
Example . For any givenk∈N, let us write the Bleimannet al.[] (in short, BBH) operators of two variables as below:
k(h;x,y) :=
( +x)k( +y)k
k
i=
k
u= h
i k–i+ ,
u k–u+
k i
k u
xiyu, ()
whereh∈Hω(D) andD= [,∞)×[,∞). We know that
( +x)k=
k
i=
k i
xi. ()
By considering the test functionh(x,y) = and solving () and (), one obtains
k(h;x,y)→ =h(x,y).
Again by solving () and () and takingh(x,y) =+xx, we get
k(h;x,y) =
( +x)k
k
i= i k+
k i
xi,
= x ( +x)k
k k+
k–
i=
k–
i
xi
= x ( +x)
and consequently
k(h;x,y)→ x
+x=h(x,y).
Similarly, for the test functionshandh, we obtain
k(h;x,y) = y
( +y)
k k+ →
y
+y=h(x,y)
and
k(h;x,y) =
x
+x
k(k– ) (k+ ) +
x
+x k
(k+ ) +
y
+y
k(k– ) (k+ ) +
x
+x k
(k+ ) →
x
+x
+
y
+y
=h(x,y).
We now suppose thatAis a (C, )-matrix, the sequences= (sk) is the same as taken in
Example ., andqn= for alln= , , . . . ,m. ThenS-limANm¯(s) = butsis not convergent.
Also we have the sequence of operators∗k:Hω∗(D)→CB(D) such that
∗k(h;x,y) = ( +sk)k(h;x,y).
We see that the sequence (∗k) satisfies the conditions ()-() of Theorem . and conse-quently, we obtain
S-lim
Qm m
j=
∞
k=
qjaj,k∗k(h;x,y) –h(x,y)
CB(D)
= .
But, on the other hand, Theorem . does not hold for (∗k), since the sequences= (sk)
(and so the sequence of operators∗k) is not convergent. Therefore, we conclude that The-orem . is stronger than TheThe-orem ..
Competing interests
The author declares that he has no competing interests.
Acknowledgements
This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. (130-695-D1435). The author, therefore, gratefully acknowledges the DSR technical and financial support.
Received: 20 December 2015 Accepted: 5 March 2016
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