# Korovkin Second Theorem viaB-Statistical A-Summability

Authors: M. Mursaleen; A. Kiliçman
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Korovkin type approximation theorems are useful tools to check whether a given sequence(Ln)n≥1 of positive linear operators on C[0,1] of all continuous functions on the real interval [0,1] is an approximation process. That is, these theorems exhibit a variety of test functions which assure that the approximation property holds on the whole space if it holds for them. Such a property was discovered by Korovkin in 1953 for the functions 1, x, and x2 in the space C[0,1] as well as for the functions 1, cos, and sin in the space of all continuous 2π-periodic functions on the real line. In this paper, we use the notion of B-statistical A-summability to prove the Korovkin second approximation theorem. We also study the rate of B-statistical A-summability of a sequence of positive linear operators defined from C2π(ℝ) into C2π(ℝ).


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## Body

## 1. Introduction and Preliminaries

Letℕ be the set of all natural numbers, K⊆ℕ, and Kn={k≤n:k∈K}. Then the natural density of K is defined by
(1)δ(K)=limn1n|Kn|=limn(C1χK)n,
if the limit exists, where the vertical bars indicate the number of elements in the enclosed set, C1=(C,1) is the Cesàro matrix of order 1, and χK denotes the characteristic sequence of K given by
(2)(χK)i={0,ifi∉K,1,ifi∈K.A sequencex=(xk) is said to be statistically convergent to L if for every ε>0, the set Kε:={k∈ℕ:|xk-L|≥ε} has natural density zero (cf. Fast [1]); that is, for each ε>0,
(3)limn1n|{k≤n:|xk-L|≥ε}|=0.
In this case, we write L=st-limx. By the symbol st we denote the set of all statistically convergent sequences. Statistical convergence of double sequences is studied in [2, 3].A matrixA=(ank)n,k=0∞ is called regular if it transforms a convergent sequence into a convergent sequence leaving the limit invariant. The well-known necessary and sufficient conditions (Silverman-Toeplitz) for A to be regular are(i)
||A||=supn∑k|ank|<∞;(ii)
limnank=0, for each k;(iii)
limn∑k‍ank=1.Freedmann and Sember [4] generalized the natural density by replacing C1 with an arbitrary nonnegative regular matrix A. A subset K of ℕ has A-density if
(4)δA(K)=limn∑k∈K‍ank
exists. Connor [5] and Kolk [6] extended the idea of statistical convergence to A-statistical convergence by using the notion of  A-density.A sequencex is said to be A-statistically convergent to L if δA(Kϵ)=0 for every ϵ>0. In this case we write stA-limxk=L. By the symbol stA we denote the set of all A-statistically convergent sequences.In [7], Edely and Mursaleen generalized these statistical summability methods by defining the statistical A-summability and studied its relationship with A-statistical convergence.LetA=(aij) be a nonnegative regular matrix. A sequence x is said to be statistically A-summable to L if for every ϵ>0, δ({i≤n:|yi-L|≥ϵ})=0; that is,
(5)limn1n|{i≤n:|yi-L|≥ϵ}|=0,
where yi=Ai(x). Thus x is statistically A-summable to L if and only if Ax is statistically convergent to L. In this case we write L=(A)st-limx=st-limAx. By (A)st we denote the set of all statistically A-summable sequences. A more general case of statistically A-summability is discussed in [8].Quite recently, Edely [9] defined the concept of B-statistical A-summability for nonnegative regular matrices A and B which generalizes all the variants and generalizations of statistical convergence, for example, lacunary statistical convergence [10], λ-statistical convergence [11], A-statistical convergence [6], statistical A-summability [7], statistical (C,1)-summability [12], statistical (H,1)-summability [13], statistical (N-,p)-summability [14], and so forth.LetA=(aij) and B=(bnk) be two nonnegative regular matrices. A sequence x=(xk) of real numbers is said to be B-statisticallyA-summable to L if for every ϵ>0, the set K(ϵ)={i:|yi-L|≥ϵ} has B-density zero, thus
(6)δB(K(ϵ))=limn∑‍k∈K(ϵ)bnk=limn(BχK(ϵ))=limn∑‍kbnkχK(ϵ)(k)=0,
where yi=Ai(x)=∑jaijxj. In this case we denote by L=(A)stB-limx=stB-limAx. The set of all B-statistically A-summable sequences will be denoted by (A)stB.Remark 1.
(1) IfA=I (unit matrix), then (A)stB is reduced to the set of B-statistically convergent sequences which can be further reduced to lacunary statistical convergence and λ-statistical convergence for particular choice of the matrix B.
(2) IfB=(C,1) matrix,  then (A)stB is reduced to the set of statistically A-summable sequences.
(3) IfA=B=(C,1) matrix,  then (A)stB is reduced to the set of statistically (C,1)-summable sequences.
(4) IfB=(C,1) matrix and A=(ajk) are defined by
(7)ajk={pkPjif0≤k≤j,0otherwise,
then (A)stB is reduced to the set of statistically (N-,p)-summable sequences, where p=(pk) is a sequence of nonnegative numbers, such that p0>0 and
(8)Pj=∑k=0j‍pk⟶∞(j⟶∞).
(5) IfB=(C,1) matrix and A=(ajk) are defined by
(9)ajk={1kljif0≤k≤j,0otherwise,
where lj=∑k=0j‍(1/(k+1)), then (A)stB is reduced to the set of statistically (H,1)-summable sequences.
(6) If a sequence is convergent, then it isB-statistically A-summable, since Ax converges and has B-density zero, but not conversely.
(7) The spacesst,stB,(A)st, and (A)stB are not comparable, even if A=B(≠(C,1)).
(8) If a sequence isA-summable, then it is B-statistically A-summable.
(9) If a sequence is bounded andA-statistically convergent, then it is A-summable and hence statistically A-summable ([7], see Theorem 2.1) and B-statistically A-summable but not conversely.Example 2.
(1) Let us defineA=(aij), B=(bnk), and x=(xk) by
(10)aij={1;ifj=i2,0;otherwise,bnk=(10000········1201200000·····13013013000·····140140140140000··1501501501501500····························),xk={1;ifkisodd,0;ifkiseven.
Then
(11)∑j=1∞aijxj={1;ifiisodd,0;ifiiseven.
Herex∉st, x∉(A)st, x∉stA, and x∉(A)stA, but x is B-statistically A-summable to 1, since δB{i:|yi-1|≥ϵ}=0. On the other hand we can see that x is B-summable and hence x is B-statistically B-summable, A-statistically B-summable, B-statistically convergent, and statistically B-summable.LetF(ℝ) denote the linear space of all real-valued functions defined on ℝ. Let C(ℝ) be the space of all functions f continuous on ℝ. We know that C(ℝ) is a Banach space with norm
(12)∥f∥∞:=supx∈ℝ|f(x)|,f∈C(ℝ).We denote byC2π(ℝ) the space of all 2π-periodic functions f∈C(ℝ) which is a Banach space with
(13)∥f∥2π=supt∈ℝ|f(t)|.The classical Korovkin first and second theorems statewhatfollows [15, 16]:Theorem I.
Let(Tn) be a sequence of positive linear operators from C[0,1] into F[0,1]. Then limn∥Tn(f,x)-f(x)∥∞=0, for all f∈C[0,1] if and only if   limn∥Tn(fi,x)-ei(x)∥∞=0, for i=0,1,2, where e0(x)=1,e1(x)=x, and e2(x)=x2.Theorem II.
Let(Tn) be a sequence of positive linear operators from C2π(ℝ) into F(ℝ). Then limn∥Tn(f,x)-f(x)∥∞=0,  for all f∈C2π(ℝ) if and only if   limn∥Tn(fi,x)-fi(x)∥∞=0,  for i=0,1,2, where f0(x)=1,f1(x)=cosx, and f2(x)=sinx.We writeLn(f;x) for Ln(f(s);x), and we say that L is a positive operator if L(f;x)≥0 for all f(x)≥0.The following result was studied by Duman [17] which is A-statistical analogue of Theorem II.Theorem A.
LetA=(ank) be a nonnegative regular matrix, and let (Tk) be a sequence of positive linear operators from C2π(ℝ) into C2π(ℝ). Then for all f∈C2π(ℝ)(14)stA-limk→∞∥Tk(f;x)-f(x)∥2π=0
if and only if
(15)stA-limk→∞∥Tk(1;x)-1∥2π=0,stA-limk→∞∥Tk(cost;x)-cosx∥2π=0,stA-limk→∞∥Tk(sint;x)-sinx∥2π=0.Recently, Karakuş and Demirci [18] proved Theorem II for statistical A-summability.Theorem B.
LetA=(ank) be a nonnegative regular matrix, and let (Tk) be a sequence of positive linear operators from C2π(ℝ) into C2π(ℝ). Then for all f∈C2π(ℝ)(16)st-limn→∞∥∑k‍ankTk(f;x)-f(x)∥2π=0
if and only if
(17)st-limn→∞∥∑k‍ankTk(1;x)-1∥2π=0,st-limn→∞∥∑k‍ankTk(cost;x)-cosx∥2π=0,st-limn→∞∥∑k‍ankTk(sint;x)-sinx∥2π=0.Several mathematicians have worked on extending or generalizing the Korovkin's theorems in many ways and to several settings, including function spaces, abstract Banach lattices, Banach algebras, and Banach spaces. This theory is very useful in real analysis, functional analysis, harmonic analysis, measure theory, probability theory, summability theory, and partial differential equations. But the foremost applications are concerned with constructive approximation theory which uses it as a valuable tool. Even today, the development of Korovkin-type approximation theory is far frombeingcomplete. Note that the first and the second theorems of Korovkin are actually equivalent to the algebraic and the trigonometric version, respectively, of the classical Weierstrass approximation theorem [19]. Recently, such type of approximation theorems has been proved by many authors by using the concept of statistical convergence and its variants, for example, [20–28]. Further Korovkin type approximation theorems for functions of two variables are proved in [29–32]. In [29, 33] authors have used the concept of almost convergence. In this paper, we prove Korovkin second theorem by applying the notion of B-statistical A-summability. We give here an example to justify that our result is stronger than Theorems II, A, and B. We also study the rate of B-statistical A-summability of a sequence of positive linear operators defined from C2π(ℝ) into C2π(ℝ).


## 2. Main Result

Now, we prove Theorem II forB-statistically A-summability.Theorem 3.
LetA=(ank) and B=(bnk) be nonnegative regular matrices, and let (Tk) be a sequence of positive linear operators from C2π(ℝ) into C2π(ℝ). Then for all f∈C2π(ℝ)(18)stB-limn→∞∥∑k‍ankTk(f;x)-f(x)∥2π=0
if and only if
(19)stB-limn→∞∥∑k‍ankTk(1;x)-1∥2π=0,stB-limn→∞∥∑k‍ankTk(cost;x)-cosx∥2π=0,stB-limn→∞∥∑k‍ankTk(sint;x)-sinx∥2π=0.Proof.
Since each of 1,cosx, and sinx belongs to C2π(ℝ), conditions (19) follow immediately from (18).  Let the conditions (19) hold and f∈C2π(ℝ). Let I  be a closed subinterval of length 2π of   ℝ. Fix x∈I. By the continuity of f at x, it follows that for given ε>0 there is a number δ>0, such that for all t(20)|f(t)-f(x)|<ε,
whenever |t-x|<δ. Since f is bounded, it follows that
(21)|f(t)-f(x)|≤2∥f∥2π,
for all t∈ℝ. For all t∈(x-δ,2π+x-δ], it is well known that
(22)|f(t)-f(x)|<ε+2∥f∥2πsin2(δ/2)ψ(t),
where ψ(t)=sin2((t-x)/2). Since the function f∈C2π(ℝ) is 2π-periodic, the inequality (22) holds for t∈ℝ.
Now, operatingTk(1;x) to this inequality, we obtain
(23)|Tk(f;x)-f(x)|≤(ε+|f(x)|)|Tk(1;x)-1|+ε+∥f∥2πsin2(δ/2){|Tk(1;x)-1|+|cosx|+∥f∥2πsin2δ2×|Tk(cost;x)-cosx|+∥f∥2πsin2δ2+|sinx||Tk(sint;x)-sinx|}≤ε+(ε+|f(x)|+∥f∥2πsin2(δ/2))×{|Tk(1;x)-1|+|Tk(cost;x)-cosx|×+|Tk(sint;x)-sinx|}.
Now, taking supx∈I, we get
(24)∥Tk(f;x)-f(x)∥∞≤ε+K(∥Tk(1;x)-1∥2π+∥Tk(cost;x)-cosx∥2πε+K+∥Tk(sint;x)-sinx∥2π),
where K:=ε+∥f∥2π+(∥f∥2π/sin2(δ/2)). Now replace Tk(·,x) by ∑k‍amkTk(·,x) and then by Bm(·,x) in (24) on both sides. For a given r>0 choose ε′>0, such that ε′<r. Define the following sets
(25)D={m≤n:∥Bm(f,x)-f(x)∥2π≥r},D1={m≤n:∥Bm(f1,x)-f0∥2π≥r-ε′3K},D2={m≤n:∥Bm(f2,x)-f1∥2π≥r-ε′3K},D3={m≤n:∥Bm(f3,x)-f2∥2π≥r-ε′3K}.
Then D⊂D1∪D2∪D3, and so δB(D)≤δB(D1)+δB(D2)+δB(D3). Therefore, using conditions (19) we get (18).
This completes the proof of the theorem.


## 3. Rate ofB-Statistical A-Summability

In this section, we study the rate ofB-statistical A-summability of a sequence of positive linear operators defined from C2π(ℝ) into C2π(ℝ).Definition 4.
LetA=(aij) and B=(bnk) be two nonnegative regular matrices. Let (αn) be a positive nonincreasing sequence. We say that the sequence x=(xk) is B-statistically A-summable to the number L with the rate o(αn) if for every ε>0,
(26)limn1αn∑‍k∈K(ϵ)bnk=0,
where K(ϵ)={i:|yi-L|≥ϵ} and yi=Ai(x)=∑jaijxj as described above. In this case, we write xk-L=(A)stB-o(αn).As usual we have the following auxiliary result whose proof is standard.Lemma 5.
Let(αn) and (βn) be two positive nonincreasing sequences. Let x=(xk) and y=(yk) be two sequences, such that xk-L1=(A)stB-o(αn) and yk-L2=(A)stB-o(βn). Then(i)
c(xk-L1)=(A)stB-o(αn), for any scalar c,(ii)
(xk-L1)±(yk-L2)=(A)stB-o(γn),(iii)
(xk-L1)(yk-L2)=(A)stB-o(αnβn), where γn=max{αn,bn}.Now, we recall the notion of modulus of continuity. The modulus of continuity off∈C2π(ℝ), denoted by ω(f,δ), is defined by
(27)ω(f,δ)=sup|x-y|<δ|f(x)-f(y)|.It is well known that(28)|f(x)-f(y)|≤ω(f,δ)(|x-y|δ+1).Then prove the following result.Theorem 6.
Let(Tk) be a sequence of positive linear operators from C2π(ℝ) into C2π(ℝ).  Suppose that(i)
∥Tk(1;x)-1∥2π=(A)stB-o(αn),(ii)
ω(f,λk)=(A)stB-o(βn), where  λk=Tk(φx;x)  and  φx(y)=sin2((y-x)/2).Then for all f∈C2π(ℝ), we have
(29)∥Tk(f;x)-f(x)∥2π=(A)stB-o(γn),whereγn=max{αn,βn}.Proof.
Letf∈C2π(ℝ) and x∈[-π,π]. Using (28), we have
(30)|Tk(f;x)-f(x)|≤Tk(|f(y)-f(x)|;x)+|f(x)||Tk(1;x)-1|≤Tk(|x-y|δ+1;x)ω(f,δ)+|f(x)||Tk(1;x)-1|≤Tk(1+π2δ2sin2(y-x2);x)ω(f,δ)+|f(x)||Tk(1;x)-1|≤(Tk(1;x)+π2δ2Tk(φx;x))ω(f,δ)+|f(x)||Tk(1;x)-1|.
Put δ=λk=Tk(φx;x). Hence we get
(31)∥Tk(f;x)-f(x)∥2π≤∥f∥2π∥Tk(1;x)-1∥2π+(1+π2)ω(f,λk)+ω(f,λk)∥Tk(1;x)-1∥2π≤K{∥Tk(1;x)-1∥2π+ω(f,λk)≤K+ω(f,λk)∥Tk(1;x)-1∥2π},
where K=max{∥f∥2π,1+π2}. Hence
(32)∥Tk(f;x)-f(x)∥2π≤K{∥Tk(1;x)-1∥2π+ω(f,λk)+ω(f,λk)≤K×∥Tk(1;x)pk-1∥2π}.
Now, using Definition 4 and Conditions (i) and (ii), we get the desired result.
This completes the proof of the theorem.


## 4. Example and Concluding Remark

In the following we construct an example of a sequence of positive linear operators satisfying the conditions of Theorem3 but does not satisfy the conditions of Theorems II, A, and B.For anyn∈ℕ, denote by Sn(f) the  nth partial sum of the Fourier series of f; that is,
(33)Sn(f)(x)=12a0(f)+∑k=1n‍ak(f)coskx+bk(f)sinkx.For anyn∈ℕ, write
(34)Fn(f):=1n+1∑k=0n‍Sk(f).A standard calculation gives that for everyt∈ℝ(35)Fn(f;x)≔12π∫-ππ‍f(t)1n+1∑k=0n‍sin((2k+1)(x-t)/2)sin((x-t)/2)dt=12π∫-ππ‍f(t)1n+1∑k=0n‍sin2((n+1)(x-t)/2)sin2((x-t)/2)dt=12π∫-ππ‍f(t)φn(x-t)dt,
where
(36)φn(x):={sin2((n+1)(x-t)/2)(n+1)sin2((x-t)/2)ifxisnotamultipleof2π,n+1ifxisamultipleof2π.
The sequence (φn)n∈ℕ is a positive kernel which is called the Fejér kernel,and the corresponding operators Fn,n≥1, are called the Fejér convolution operators.We have
(37)Fn(1;x)=1,Fn(cost;x)=(nn+1)cosx,Fn(sint;x)=(nn+1)sinx.Note that the Theorems II, A, and B hold for the sequence(Fn). In fact, we have for every f∈C2π(ℝ),
(38)limn→∞Fn(f)=f.LetA=(aij),B=(bnk), and x=(xk) be defined as in Example 2. Let Ln:C2π(ℝ)→C2π(ℝ) be defined by
(39)Ln(f;x)=xnFn(f;x).
Then x is not statistically convergent, not A-statistically convergent, and not statistically A-summable, but it is B-statistically A-summable to 1. Since x is B-statistically A-summable to 1, it is easy to see that the operator Ln satisfies the conditions (19), and hence Theorem 3 holds. But on the other hand, Theorems II, A, and B do not hold for our operator defined by (39), since x (and so Ln) is not statistically convergent, not A-statistically convergent, and not statistically A-summable.Hence our Theorem3 is stronger than all the above three theorems.



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