# WeightedA-Statistical Convergence for Sequences of Positive Linear Operators

Authors: S. A. Mohiuddine; Abdullah Alotaibi; Bipan Hazarika
Journal: The Scientific World Journal
 (2014)

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## Abstract

We introduce the notion of weightedA-statistical convergence of a sequence, where A represents the nonnegative regular matrix. We also prove the Korovkin approximation theorem by using the notion of weighted A-statistical convergence. Further, we give a rate of weighted A-statistical convergence and apply the classical Bernstein polynomial to construct an illustrative example in support of our result.


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## Body

## 1. Background, Notations, and Preliminaries

We begin this paper by recalling the definition of natural (or asymptotic) density as follows. Suppose thatE⊆N:={1,2,…} and En={k≤n:k∈E}. Then
(1)δ(E)=lim⁡n1n|En|
is called thenatural density of E provided that the limit exists, where |·| represents the number of elements in the enclosed set.The term “statistical convergence” was first presented by Fast [1] which is a generalization of the concept of ordinary convergence. Actually, a root of the notion of statistical convergence can be detected by Zygmund [2] (also, see [3]), where he used the term “almost convergence” which turned out to be equivalent to the concept of statistical convergence. The notion of Fast was further investigated by Schoenberg [4], Šalát [5], Fridy [6], and Conner [7].The following notion is due to Fast [1]. A sequence x=(xk) is said to bestatistically convergent to L if δ(Kϵ)=0 for every ϵ>0, where
(2)Kϵ:={k∈N:|xk-L|≥ϵ}.
Equivalently,
(3)lim⁡nn-1|{k≤n:|xk-L|≥ϵ}|=0.
In symbol, we will write S-limx=L. We remark that every convergent sequence is statistically convergent but not conversely.LetX and Y be two sequence spaces and let A=(an,k) be an infinite matrix. If for each x=(xk) in X the series
(4)Anx=∑k‍an,kxk=∑k=1∞‍an,kxk
converges for each n∈N and the sequence Ax=(Anx) belongs to Y, then we say that matrix A maps X into Y. By the symbol (X,Y) we denote the set of all matrices which map X into Y.A matrixA (or a matrix map A) is called regular if A∈(c,c), where the symbol c denotes the spaces of all convergent sequences and
(5)lim⁡nAnx=lim⁡kxk
for all x∈c. The well-known Silverman-Toeplitz theorem (see [8]) asserts that A=(an,k) is regular if and only if(i)
lim⁡nan,k=0 for each k;(ii)
lim⁡n∑k‍an,k=1;(iii)
sup⁡n∑k‍|an,k|<∞.Kolk [9] extended the definition of statistical convergence with the help of nonnegative regular matrix A=(an,k) calling it A-statistical convergence. The definition of A-statistical convergence is given by Kolk as follows. For any nonnegative regular matrix A, we say that a sequence is A-statistically convergent to L provided that for every ϵ>0 we have
(6)lim⁡n∑k:|xn-L|≥ϵ‍an,k=L.In 2009, the concept of weighted statistical convergence was defined and studied by Karakaya and Chishti [10] and further modified by Mursaleen et al. [11] in 2012. In 2013, Belen and Mohiuddine [12] presented a generalization of this notion through de la Vallée-Poussin mean. Quite recently, Esi [13] defined and studied the notion statistical summability through de la Vallée-Poussin mean in probabilistic normed spaces.Letp=(pk) be a sequence of nonnegative numbers such that p0>0 and
(7)Pn=∑k=0n‍pk⟶∞asn⟶∞.
Let
(8)tn=1Pn∑k=0n‍pkxk,n=0,1,2….
We say that the sequence x=(xk) is (N¯,pn)-summable to L if lim⁡n→∞tn=L.The lower and upper weighted densities ofE⊆N are defined by
(9)δ_N¯(E)=liminf⁡n1Pn|{k≤Pn:k∈E}|,(10)δ¯N¯(E)=limsup⁡n1Pn|{k≤Pn:k∈E}|,
respectively. We say that E hasweighted density, denoted by δN¯(E), if the limits of both of the above densities exist and are equal; that is, one writes
(11)δN¯(E)=lim⁡n1Pn|{k≤Pn:k∈E}|.The sequencex=(xk) is said to beweighted statistically convergent (or SN¯-convergent) to L if, for every ϵ>0, the set {k∈N:pk|xk-L|≥ϵ} has weighted density zero; that is,
(12)lim⁡n1Pn|{k≤Pn:pk|xk-L|≥ϵ}|=0.
In this case we write L=SN¯-limx.Remark 1.
Ifpk=1 for all k, then (N¯,pn)-summable is reduced to (C,1)-summable (or Cesàro summable) and weighted statistical convergence is reduced to statistical convergence.On the other hand, let us recall thatC[a,b] is the space of all functions f continuous on [a,b]. We know that C[a,b] is a Banach space with norm
(13)∥f∥∞:=sup⁡x∈[a,b]|f(x)|,f∈C[a,b].
Suppose that L is a linear operator from C[a,b] into C[a,b]. It is clear that if f≥0 implies Lf≥0, then the linear operator L is positive on C[a,b]. We denote the value of Lf at a point x∈[a,b] by L(f;x). The classical Korovkin approximation theorem states the following [14].Theorem 2.
Let(Tn) be a sequence of positive linear operators from C[a,b] into C[a,b]. Then,
(14)lim⁡n∥Tn(f,x)-f(x)∥∞=0,
for all f∈C[a,b] if and only if
(15)lim⁡n∥Tn(fi,x)-fi(x)∥∞=0,
where fi(x)=xi and i=0,1,2.Many mathematicians extended the Korovkin-type approximation theorems by using various test functions in several setups, including Banach spaces, abstract Banach lattices, function spaces, and Banach algebras. Firstly, Gadjiev and Orhan [15] established classical Korovkin theorem through statistical convergence and display an interesting example in support of our result. Recently, Korovkin-type theorems have been obtained by Mohiuddine [16] for almost convergence. Korovkin-type theorems were also obtained in [17] for λ-statistical convergence. The authors of [18] established these types of approximation theorem in weighted Lp spaces, where 1≤p<∞, through A-summability which is stronger than ordinary convergence. For these types of approximation theorems and related concepts, one can be referred to [19–27] and references therein.


## 2. Korovkin-Type Theorems by WeightedA-Statistical Convergence

Kolk [9] introduced the notion of A-statistical convergence by considering nonnegative regular matrix A instead of Cesáro matrix in the definition of statistical convergence due to Fast. Inspired from the work of Kolk, we introduce the notion of weighted A-statistical convergence of a sequence and then we establish some Korovkin-type theorems by using this notion.Definition 3.
LetA=(an,k) be a nonnegative regular matrix. A sequence x=(xk) of real or complex numbers is said to beweightedA-statistically convergent, denoted by SAN¯-convergent, to L if for every ϵ>0(16)lim⁡n∑k∈E(p,ϵ)‍an,k=0,
where
(17)E(p,ϵ)={k∈N:pk|xk-L|≥ϵ}.
In symbol, we will write SAN¯-limx=L.Remark 4.
One has the following.(i)
If we takeA=I, where I denotes the identity matrix, then weighted A-statistical convergence of a sequence is reduced to ordinary convergence.(ii)
If we takeA=(C,1), where (C,1) denotes the Cesáro matrix of order one, then weighted A-statistical convergence of a sequence reduces to weighted statistical convergence.(iii)
If we takeA=(C,1) and pk=1 for all k, then weighted A-statistical convergence of a sequence reduces to statistical convergence.Note that convergent sequence implies weightedA-statistically convergent to the same value but the converse is not true in general. For example, take A=(C,1) and pk=1 for all k and define a sequence x=(xk) by
(18)xk={1,ifk=n2,0,otherwise,
where n∈N. Then this sequence is statistically convergent to 0 but not convergent; in this case, weighted A-statistical convergence of a sequence coincides with statistical convergence.Theorem 5.
LetA=(an,k) be a nonnegative regular matrix. Consider a sequence of positive linear operators (Mk) from C[a,b] into itself. Then, for all f∈C[a,b] bounded on whole real line,
(19)SAN¯-lim⁡k∥Mk(f,x)-f(x)∥∞=0
if and only if
(20)SAN¯-lim⁡k∥Mk(1,x)-1∥∞=0,SAN¯-lim⁡k∥Mk(v,x)-x∥∞=0,SAN¯-lim⁡k∥Mk(v2,x)-x2∥∞=0.Proof.
Equation (20) directly follows from (19) because each of 1,x,x2 belongs to C[a,b]. Consider a function f∈C[a,b]. Then there is a constant C>0 such that |f(x)|≤C for all x∈(-∞,∞). Therefore,
(21)|f(v)-f(x)|≤2C,-∞<v,x<∞.
Let ϵ>0 be given. By hypothesis there is a δ:=δ(ɛ)>0 such that
(22)|f(v)-f(x)|<ϵ∀|v-x|<δ.
Solving (21) and (22) and then substituting Ω(v)=(v-x)2, one obtains
(23)|f(v)-f(x)|<ϵ+2Cδ2Ω,∀|v-x|<δ.
Equation (23) can also be written as
(24)-ϵ-2Cδ2Ω<f(v)-f(x)<ϵ+2Mδ2Ω.
Operating Mk(1,x) to (24) since Mk(f,x) is linear and monotone, one obtains
(25)Mk(1,x)(-ϵ-2Cδ2Ω)<Mk(1,x)(f(v)-f(x))<Mk(1,x)(ϵ+2Cδ2Ω).
Note that x is fixed, so f(x) is constant number. Thus, we obtain from (25) that
(26)-ϵMk(1,x)-2Cδ2Mk(Ω,x)<Mk(f,x)-f(x)Mk(1,x)<ϵMk(1,x)+2Cδ2Mk(Ω,x).
The term “Mk(f,x)-f(x)Mk(1,x)” in (26) can also be written as
(27)Mk(f,x)-f(x)Mk(1,x)=Mk(f,x)-f(x)-f(x)[Mk(1,x)-1].
Now substituting the value of Mk(f,x)-f(x)Mk(1,x) in (26), we get that
(28)Mk(f,x)-f(x)<ϵMk(1,x)+2Cδ2Mk(Ω,x)+f(x)(Mk(1,x)-1).
We can rewrite the term “Mk(Ω,x)” in (28) as follows:
(29)Mk(Ω,x)=Mk((v-x)2,x)=Mk(v2,x)+2xMk(v,x)+x2Mk(1,x)=[Mk(v2,x)-x2]-2x[Mk(v,x)-x]+x2[Mk(1,x)-1].
Equation (28) with the above value of Mk(Ω,x) becomes
(30)Mk(f,x)-f(x)<ϵMk(1,x)+2Cδ2{[Mk(v2,x)-x2]+2x[Mk(v,x)-x]+x2[Mk(1,x)-1]}+f(x)(Mk(1,x)-1)=ϵ[Mk(1,x)-1]+ϵ+2Cδ2{[Mk(v2,x)-x2]+2x[Mk(v,x)-x]+x2[Mk(1,x)-1]}+f(x)[Mk(1,x)-1].
Therefore,
(31)|Mk(f,x)-f(x)|≤(ϵ+2Cb2δ2+C)|Mk(1,x)-1|+2Cδ2|Mk(v2,x)-x2|+4Cbδ2|Mk(v,x)-x|,
where b=max⁡|x|. Taking supremum over x∈[a,b], one obtains
(32)∥Mk(f,x)-f(x)∥∞≤(ϵ+2Cb2δ2+C)∥Mk(1,x)-1∥∞+2Cδ2∥Mk(v2,x)-x2∥∞+4Cbδ2∥Mk(v,x)-x∥∞,
or
(33)∥Mk(f,x)-f(x)∥∞≤T{∥Mk(1,x)-1∥∞+∥Mk(v2,x)-x2∥∞+∥Mk(v,x)-x∥∞∥Mk(v2,x)-x2∥∞},
where
(34)T:=max⁡{ϵ+2Cb2δ2+C,2Cδ2,4Cbδ2}.
Hence,
(35)pk∥Mk(f,x)-f(x)∥∞≤T{pk∥Mk(1,x)-1∥∞+pk∥Mk(v2,x)-x2∥∞+pk∥Mk(v,x)-x∥∞∥Mk(v2,x)-x2∥∞}.
For a given α>0, choose ϵ>0 such that ϵ<α, and we will define the following sets:
(36)E={k∈N:pk∥Mk(f,x)-f(x)∥∞≥α},E1={k∈N:pk∥Mk(1,x)-1∥∞≥α-ϵ3T},E2={k∈N:pk∥Mk(v,x)-x∥∞≥α-ϵ3T},E3={k∈N:pk∥Mk(v2,x)-x2∥∞≥α-ϵ3T}.
It is easy to see that
(37)E⊂E1∪E2∪E3.
Thus, for each n∈N, we obtain from (35) that
(38)∑k∈E‍an,k≤∑k∈E1‍an,k+∑k∈E2‍an,k+∑k∈E3‍an,k.
Taking limit n→∞ in (38) and also (20) gives that
(39)lim⁡n∑k∈E‍an,k=0.
This yields that
(40)SAN¯-lim⁡k∥Mk(f,x)-f(x)∥∞=0,
for all f∈C[a,b].We also obtain the following Korovkin-type theorem for weighted statistical convergence by writing Cesáro matrix(C,1) instead of nonnegative regular matrix A in Theorem 5.Theorem 6.
Consider a sequence of positive linear operators(Mk) from C[a,b] into itself. Then, for all f∈C[a,b](41)SN¯-lim⁡k∥Mk(f,x)-f(x)∥∞=0
if and only if
(42)SN¯-lim⁡k∥Mk(1,x)-1∥∞=0,(43)SN¯-lim⁡k∥Mk(v,x)-x∥∞=0,(44)SN¯-lim⁡k∥Mk(v2,x)-x2∥∞=0.Proof.
Following the proof of Theorem5, one obtains
(45)E⊂E1∪E2∪E3
and so
(46)δN¯(E)⊂δN¯(E1)+δN¯(E2)+δN¯(E3).
Equations (42)–(44) give that
(47)SN¯-lim⁡k∥Mk(f,x)-f(x)∥∞=0.Remark 7.
If we replace nonnegative regular matrixA by Cesáro matrix and choose pk=1 for all k, in Theorem 5, then we obtain Theorem 1 due to Gadjiev and Orhan [15].Remark 8.
By Theorem 2 of [10], we have that if a sequence x=(xk) is weighted statistically convergent to L, then it is strongly (N¯,pn)-summable to L provided that pk|xk-L| is bounded; that is, there exists a constant C such that pk|xk-L|≤C for all k∈N. We write
(48)|N¯,pn|={x=(xk):lim⁡n1Pn∑k=0n‍pk|xk-L|=0forsomeL}
for the set of all sequences x=(xk) which are strongly (N¯,pn)-summable to L.Theorem 9.
LetMk:C[a,b]→C[a,b] be a sequence of positive linear operators which satisfies (43)-(44) of Theorem 6 and the following condition holds:
(49)lim⁡k∥Mk(1,x)-1∥∞=0.
Then,
(50)lim⁡n1Pn∑k=0n‍pk∥Mk(f,x)-f(x)∥∞=0,
for any f∈C[a,b].Proof.
It follows from (49) that ∥Mk(1,x)∥∞≤C′, for some constant C′>0 and for all k∈N. Hence, for f∈C[a,b], one obtains
(51)pk∥Mk(f,x)-f(x)∥∞≤pk(∥f∥∞∥Mk(1,x)∥∞+∥f∥∞)≤pkC(C′+1).
Right hand side of (51) is constant, so pk∥Mk(f,x)-f(x)∥∞ is bounded. Since (49) implies (42), by Theorem 6 we get that
(52)SN¯-lim∥Mk(f,x)-f(x)∥∞=0.
By Remark 8, (51) and (52) together give the desired result.


## 3. Rate of WeightedA-Statistical Convergence

First we define the rate of weightedA-statistically convergent sequence as follows.Definition 10.
LetA=(an,k) be a nonnegative regular matrix and let (ak) be a positive nonincreasing sequence. Then, a sequence x=(xk) is weighted A-statistically convergent to L with the rate of o(ak) if for each ϵ>0(53)lim⁡n1an∑k∈E(p,ϵ)‍an,k=0,
where
(54)E(p,ϵ)={k∈N:pk|xk-L|≥ϵ}.
In symbol, we will write
(55)xk-L=SAN¯-o(ak)ask⟶∞.We will prove the following auxiliary result by using the above definition.Lemma 11.
LetA=(an,k) be a nonnegative regular matrix. Suppose that (ak) and (bk) are two positive nonincreasing sequences. Let x=(xk) and y=(yk) be two sequences such that xk-L1=SAN¯-o(ak) and yk-L2=SAN¯-o(bk). Then,(i)
(xk-L1)±(yk-L2)=SAN¯-o(ck),(ii)
(xk-L1)(yk-L2)=SAN¯-o(ck),(iii)
α(xk-L1)=SAN¯-o(ak), for any scalar α,where ck=max⁡{ak,bk}.Proof.
(i) Suppose that(56)xk-L1=SAN¯-o(ak),yk-L2=SAN¯-o(bk).
Given ϵ>0, define
(57)E′={k∈N:pk|(xk-L1)±(yk-L2)|≥ϵ},E′′={k∈N:pk|xk-L1|≥ϵ2},E′′′={k∈N:pk|yk-L2|≥ϵ2}.
It is easy to see that
(58)E′⊂E′′∪E′′′.
This yields that
(59)1cn∑k∈E′‍an,k≤1cn∑k∈E′′‍an,k+1cn∑k∈E′′′‍an,k
holds for all n∈N. Since ck=max⁡{ak,bk}, (59) gives that
(60)1cn∑k∈E′‍an,k≤1an∑k∈E′′‍an,k+1bn∑k∈E′′′‍an,k.
Taking limit n→∞ in (60) together with (56), we obtain
(61)lim⁡n1cn∑k∈E′‍an,k=0.
Thus,
(62)(xk-L1)±(yk-L2)=SAN¯-o(ck).
Similarly, we can prove (ii) and (iii).Recall that the modulus of continuity off in C[a,b] is defined by
(63)ω(f,δ)=sup⁡{|f(x)-f(y)|:x,y∈[a,b],|x-y|<δ}.
It is well known that
(64)|f(x)-f(y)|≤ω(f,δ)(|x-y|δ+1).Theorem 12.
LetA=(an,k) be a nonnegative regular matrix. If the sequence of positive linear operators Mk:C[a,b]→C[a,b] satisfies the conditions(i)
∥Mk(1;x)-1∥∞=SAN¯-o(ak),(ii)
ω(f,λk)=SAN¯-o(bk) with λk=Mk(φx;x) and φx(y)=(y-x)2,where (ak) and (bk) are two positive nonincreasing sequences, then
(65)∥Mk(f;x)-f(x)∥∞=SAN¯-o(ck),
for all f∈C[a,b], where ck=max⁡{ak,bk}.Proof.
Equation (27) can be reformed into the following form:
(66)|Mk(f;x)-f(x)|≤Mk(|f(y)-f(x)|;x)+|f(x)||Mk(1;x)-1|≤Mk(1+|y-x|δ;x)ω(f,δ)+|f(x)||Mk(1;x)-1|≤Mk(1+(y-x)2δ2;x)ω(f,δ)+|f(x)||Mk(1;x)-1|≤(Mk(1;x)+1δ2Mk(φx;x))ω(f,δ)+|f(x)||Mk(1;x)-1|≤|Mk(1;x)-1|ω(f,δ)+|f(x)||Mk(1;x)-1|+ω(f,δ)+1δ2Mk(φx;x)ω(f,δ).
Choosing δ=λk=Mk(φx;x), one obtains
(67)∥Mk(f;x)-f(x)∥∞≤T∥Mk(1;x)-1∥∞+2ω(f,λk)+∥Mk(1;x)-1∥∞ω(f,λk),
where T=∥f∥∞. For a given ϵ>0, we will define the following sets:
(68)E1′={k∈N:pk∥Mk(f,x)-f(x)∥∞≥ϵ},E2′={k∈N:pk∥Mk(1,x)-1∥∞≥ϵ3T},E3′={k∈N:pkω(f,λk)≥ϵ6},E4′={k∈N:pkω(f,λk)∥Mk(1,x)-1∥∞≥ϵ3}.
It follows from (67) that
(69)1cn∑k∈E1′‍an,k≤1cn∑k∈E2′‍an,k+1cn∑k∈E3′‍an,k+1cn∑k∈E4′‍an,k
holds for n∈N. Since ck=max⁡{ak,bk}, we obtain from (69) that
(70)1cn∑k∈E1′‍an,k≤1an∑k∈E2′‍an,k+1bn∑k∈E3′‍an,k+1cn∑k∈E4′‍an,k.
Taking limit n→∞ in (70) together with Lemma 11 and our hypotheses (i) and (ii), one obtains
(71)lim⁡n1cn∑k∈E1′‍an,k=0.
This yields
(72)∥Mk(f;x)-f(x)∥∞=SAN¯-o(ck).


## 4. Example and the Concluding Remark

The operatorsBn:C[0;1]→C[0;1] given by
(73)Bn(f,x):=∑k=0n‍pn,k(x)f(kn),
where pn,k(x) are the fundamental Bernstein polynomials defined by
(74)pn,k(x)=(nk)xk(1-x)n-k
for any x∈[0,1], any k∈{0,1,…,n}, and any n∈N, are called Bernstein operators and were first introduced in [28]. Let the sequence (An) be defined by An:C[0,1]→C[0,1] with An(f,x)=(1+xn)Bn(f,x), where x=(xk) is a sequence defined by
(75)x=(xk)={k,ifk=n2,n∈N,0,otherwise.
That is, (xk)=(1,0,0,2,0,0,0,0,3,0,…,0,4,0,0,…). Let pk=k(k=1,2,…) and consider a nonnegative regular matrix A=(C,1). Then,
(76)pkxk=(1,0,0,8,0,0,0,0,27,0,…,0,64,0,0,…),Pn=∑k=1n‍pk=n(n+1)2.
Since
(77)lim⁡n1Pn|{k≤Pn:pk|xk-0|≥ϵ}|≤lim⁡n1PnPn=1n(n+1)/2=0,(xk) is weighted statistically convergent to 0 but not convergent. It is not difficult to see that
(78)Bn(1,x)=1,Bn(t,x)=x,Bn(t2,x)=x2+x-x2n
and the sequence (An) satisfies conditions (20). This yields that
(79)S(C,1)N¯-lim∥An(f,x)-f(x)∥∞=0.On the other hand, one obtainsAn(f,0)=(1+xn)f(0), since Bn(f,0)=f(0), and hence
(80)∥An(f,x)-f(x)∥∞≥|An(f,0)-f(0)|=xn|f(0)|.
It follows that (An) does not satisfy the Korovkin theorem, since (xn) and hence (An) is not convergent. Finally, we conclude that Theorem 5 is stronger than Theorem 2.



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