# Generalization of Statistical Korovkin Theorems

Authors: Alperen Ali Ergur; Oktay Duman
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We generalize and develop the Korovkin-type approximation theory by using an appropriate abstract space. We show that our approximation is more applicable than the classical one. At the end, we
display some applications.


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## Body

## 1. Introduction

The classical Korovkin theory enables us to approximate a function by means of positive linear operators (see, e.g., [1–3]). In recent years, this theory has been quite improved by some efficient tools in mathematics such as the concept of statistical convergence from summability theory, the fuzzy logic theory, the complex functions theory, the theory of q-calculus, and the theory of fractional analysis. The main purpose of this paper is to generalize and develop this Korovkin theory by using an appropriate abstract space. Actually, the most important motivation of this study has its roots from the paper by Yoshinaga and Tamura [4]. In the present paper, we show that our new approximation is more general and also more applicable than that of [4].Throughout the paper the following assumptions are imposed:(i)
(X,𝒰) is a Hausdorff uniform space provided with the uniform structure (X,𝒰);(ii)
𝒰 is the filter of the surroundings containing the diagonal Δ={(x,x):x∈X} in X×X;(iii)
ℱ is a vector space of real-valued functions defined on X including the constant-valued function e0(x)=1;(iv)
Y is a compact subspace of X;(v)
Ln is a positive linear operator of ℱ into ℝY for each n∈ℕ;(vi)
A:=[ajn] is a nonnegative regular summable matrix.Assume further that there exists a certain real-valued functionF(x,y) satisfying the following conditions:(i)
F(x,y)≥0 on X×Y and F(y,y)=0 for each y∈Y;(ii)
Fy∈ℱ for each y∈Y, where Fy is the function on X defined by Fy(x):=F(x,y);(iii)
for eachy∈Y,  Fy(x) is continuous with respect to x at each point in Y;(iv)
ρ(U)=infF(x,y)>0 for each U∈𝒰, where the infimum is taken over (X×Y)-U;(v)
there existy1,y2∈Y, y1≠y2, such that Fy1(x) and Fy2(x) are bounded functions of x, and it holds that
(1)stA-limn∥Ln(Fy1)-Fy1∥=0,stA-limn∥Ln(Fy2)-Fy2∥=0,stA-limn(supy∈YLn(Fy;y))=0,where the symbol∥·∥ denotes the classical sup-norm on the compact set Y. Here, we use the concept of A-statistical convergence, where A is a nonnegative regular summable matrix. Recall that, for a given subset K of ℕ, the A-density of K, denoted by δA(K), is defined to be δA(K)=limj→∞∑n∈K‍ajn provided that the limit exists. Using this A-density, we say that a sequence x=(xn) is A-statistically convergent to L if and only if δA(K(ε))=0 for every ε>0, where K(ε):={n∈ℕ:|xn-L|≥ε} (see [5]). In the case of A=C1, the Cesàro matrix, it reduces to the concept of statistical convergence introduced by Fast [6]. Of course, if we take A=I, the identity matrix, then we get the ordinary convergence.We should note that if, for eachy∈Y, Fy(x) is a bounded function of x for which
(2)stA-limn(supx,y∈Y|Ln(Fy;x)-Fy(x)|)=0holds, then we get the conditions in(v).Then, with the above terminology, Yoshinaga and Tamura [4] proved the following approximation result (in the case of A=I).Theorem A (see [4]).
Letf be a bounded real-valued function on X and continuous at each point in Y. Then, if f∈ℱ, the sequence {Ln(f)} is uniformly convergent to f on Y.


## 2. Statistical Approximation Theorem

In this section, we obtain the statistical analog of Theorem A in order to get a more applicable approximation theorem.We first need the following three lemmas.Lemma 1 (see [4]).
LetV be an open subset of X×Y containing ΔY:=Δ∩(X×Y). Then, it is possible to see that U∩(X×Y)⊂V for some U∈𝒰.Lemma 2.
The sequence{∥Ln(e0)∥} is A-statistically bounded; that is; there exist a positive real number M and a subset K⊂ℕ having A-density 1 such that
(3)∥Ln(e0)∥≤Mforeveryn∈K.Proof.
For the pointsy1,y2 given in (v), we can take U0∈𝒰 so that (y1,y2)∉U0. Now, choose U∈𝒰 such that U=U-1 and U∘U⊂U0. Then, we observe, for every x∈X, that
(4)Fy1(x)+Fy2(x)≥ρ(U).
Hence, we get, for eachy∈Y and for every n∈ℕ, that
(5)Ln(Fy1;y)+Ln(Fy2;y)≥ρ(U)Ln(e0;y)≥0,
which implies that(6)ρ(U)Ln(e0;y)≤|Ln(Fy1;y)-Fy1(y)|ρ(U)Ln(e0;y)≤+|Ln(Fy2;y)-Fy2(y)|ρ(U)Ln(e0;y)≤+{Fy1(y)+Fy2(y)}.
Taking supremum overy∈Y and also letting
(7)M1≔1ρ(U)∥Fy1+Fy2∥,
we obtain, for everyn∈ℕ, that
(8)∥Ln(e0)∥≤1ρ(U)∥Ln(Fy1)-Fy1∥∥Ln(e0)∥≤+1ρ(U)∥Ln(Fy2)-Fy2∥+M1.
Now, for a givenε>0, define the following sets:
(9)K1≔{n∈ℕ:∥Ln(Fy1)-Fy1∥≥ερ(U)2},K2≔{n∈ℕ:∥Ln(Fy2)-Fy2∥≥ερ(U)2}.
Then, from the conditions in(v), we may write that
(10)δA(K1)=δA(K2)=0.
Now setting(11)K≔ℕ-(K1∪K2),
we immediately get that(12)δA(K)=1.
Furthermore, it follows from (8) that, for every n∈K, that is, n∉K1 and n∉K2,
(13)∥Ln(e0)∥≤ε+M1=:M,
which completes the proof.Lemma 3.
Letg(x,y) be a real-valued and bounded function on X×Y, and let g be continuous at each diagonal point (y,y)∈ΔY. For each y∈Y, define the function gy on X by gy(x):=g(x,y). Assume further that gy∈ℱ and gy(y)=0 for each y∈Y. Then, one has
(14)stA-limn(supy∈Y|Ln(gy;y)|)=0.Proof.
Sinceg is continuous at any diagonal point (y,y)∈ΔY, for every ε>0, there exists an open neighborhood V(y) of y in X such that
(15)|g(x,y)|=|g(x,y)-g(y,y)|<ε
for every(x,y)∈V(y)×(V(y)∩Y). Now, if we define the set V by
(16)V≔⋃y∈Y‍{V(y)×(V(y)∩Y)},
then we easily see thatV is an open subset of X×Y containing the diagonal ΔY. Also, it follows from Lemma 1 that U∩(X×Y)⊂V for some U∈𝒰. Now, setting
(17)C≔sup(x,y)∈X×Y|g(x,y)|,
we get, for every(x,y)∈X×Y, that
(18)|g(x,y)|≤ε+Cρ(U)Fy(x),
which in turn implies that(19)|Ln(gy;y)|≤Ln(|gy|;y)|Ln(gy;y)|≤εLn(e0;y)+Cρ(U)Ln(Fy;y).
Thus we conclude that, for everyn∈ℕ, the inequality
(20)supy∈Y|Ln(gy;y)|≤ε∥Ln(e0)∥+Cρ(U)supy∈YLn(Fy;y)
holds. By Lemma2, there exists a positive real number M and a subset K of ℕ having A-density 1 such that
(21)supy∈Y|Ln(gy;y)|≤Mε+Cρ(U)supy∈YLn(Fy;y)
holds for everyn∈K. Now, for a given r>0, choose an ε>0 such that r>Mε. Then, considering the following subsets of ℕ:
(22)D≔{n∈ℕ:supy∈Y|Ln(gy;y)|≥r},D′≔{n∈ℕ:supy∈YLn(Fy;y)≥(r-εM)ρ(U)C}
and also using (21), we have D∩K⊂D′∩K⊂D′, which gives, for every j∈ℕ, that
(23)0≤∑n∈D∩K‍ajn≤∑n∈D′∩K‍ajn≤∑n∈D′‍ajn.
Taking limit asj→∞ in both sides of the last inequality and also using (v), we obtain that
(24)limj∑n∈D∩K‍ajn=0.
Furthermore, we may write that(25)∑n∈D‍ajn=∑n∈D∩K‍ajn+∑n∈D∩(ℕ-K)‍ajn∑n∈D‍ajn≤∑n∈D∩K‍ajn+∑n∈(ℕ-K)‍ajn.
SinceδA(K)=1, we get δA(ℕ-K)=0. Thus, by (24) and (25), we obtain that
(26)limj∑n∈D‍ajn=0,
which means(27)stA-limn(supy∈Y|Ln(gy;y)|)=0.
Therefore, the proof is completed.Now we are ready to give our main approximation result in statistical sense.Theorem 4.
Letf be a bounded real-valued function on X and continuous at each point in Y. Then, if f∈ℱ, one has
(28)stA-limn∥Ln(f)-f∥=0.Proof.
As in the proof of Lemma2, we take U0,   U∈𝒰 such that U=U-1,  (y1,y2)∉U0, and U∘U⊂U0, where y1,y2 are given in (v). Let
(29)G(x)≔F(x,y1)+F(x,y2),g(x,y)≔f(x)-f(y)G(y)G(x)
for(x,y)∈X×Y. Then, we see that G∈ℱ, and so 0<ρ(U)≤G(x)≤C for some C>0 due to the boundedness of the functions Fy1 and Fy2 on X. Also, observe that gy∈ℱ and gy(y)=0 for each y∈Y. Since f and G are continuous at any point y∈Y and also G(y)≥ρ(U)>0, we easily check that the function g(x,y) is continuous at each point (y,y)∈Δy. Since f is bounded on X, we may write that M:=supx∈X|f(x)|<∞. Then, it is not hard to see that
(30)|g(x,y)|≤M+Mρ(U)C
for every(x,y)∈X×Y. From Lemma 3, one can get that
(31)stA-limn(supy∈Y|Ln(gy;y)|)=0.
On the other hand, by (29), we have
(32)gy(x)=f(x)-f(y)Fy1(y)+Fy2(y)(Fy1(x)+Fy2(x)),
which yields that(33)Ln(gy;y)=Ln(f;y)Ln(gy;y)=-f(y)Fy1(y)+Fy2(y)(Ln(Fy1;y)+Ln(Fy2;y)).
Hence we get(34)Ln(f;y)-f(y)=Ln(gy;y)+f(y)Fy1(y)+Fy2(y)×{(Ln(Fy1;y)-Fy1(y))+(Ln(Fy2;y)-Fy2(y))}.
Taking supremum overy∈Y, we immediately obtain that
(35)∥Ln(f)-f∥≤supy∈Y|Ln(gy;y)|∥Ln(f)-f∥≤+Mρ(U){∥Ln(Fy1)-Fy1∥∥Ln(f)-f∥≤+Mρ(U)+∥Ln(Fy2)-Fy2∥}.
Now, for a givenε>0, define the following sets:
(36)E≔{n∈ℕ:∥Ln(f)-f∥≥ε},E1≔{n∈ℕ:supy∈Y|Ln(gy;y)|≥ε3},E2≔{n∈ℕ:∥Ln(Fy1)-Fy1∥≥ερ(U)3M},E3≔{n∈ℕ:∥Ln(Fy2)-Fy2∥≥ερ(U)3M}.
Then, it follows from (35) that
(37)E⊂E1∪E2∪E3,
which guarantees that, for anyj∈ℕ,
(38)∑n∈E‍ajn≤∑n∈E1‍ajn+∑n∈E2‍ajn+∑n∈E3‍ajn.
Now lettingj→∞ and also using (v) and (31), we conclude that
(39)limj∑n∈E‍ajn=0,
which is the desired result.


## 3. Concluding Remarks

If we takeA=I, the identity matrix, in Theorem 4, then we easily get Theorem A. Hence, one can say that Theorem 4 covers Theorem A. However, if we take A=C1, the Cesàro matrix, and also define the sequence (un) by
(40)un:={1,n=m2,m∈ℕ,0,otherwise,then we observe that(41)stC1-limnun=st-limnun=0although it is nonconvergent in the usual sense. Now, assume that{Ln} is a sequence of positive linear operators satisfying all conditions of Theorem A. Then, using (un) and (Ln), we construct new operators Tn as follows:
(42)Tn(f)≔(1+un)Ln(f)foreveryf∈ℱ.In this case, we verify that our operatorsTn satisfy all conditions of Theorem 4 due to property (41). Thus, we may write that, for every f∈ℱ,
(43)st-limn∥Tn(f)-f∥=0.However, since the sequence(un) given by (40) is nonconvergent, approximating a function f∈ℱ by the operators Tn is impossible. This example clearly shows that Theorem 4 is a nontrivial generalization of Theorem A.Now we give some significant applications of Theorem4. As usual, by C(X) we denote the space of all real-valued continuous functions on X.Corollary 5 (see Theorem  3.5 of [7]).
LetX be a compact Hausdorff space, and let f1,f2,…,fm∈C(X) satisfy the condition that there exist g1,g2,…,gm∈C(X) such that defining P(x,y):=∑i=1m‍gi(y)fi(x) for every x,y∈X, it holds that P(x,y)≥0 and P(x,y)=0 if and only if x=y. Assume that {Ln} is a sequence of positive linear operators from C(X) into itself. Assume further that, for a given nonnegative regular summable matrix A=[ajn],
(44)stA-limn∥Ln(fi)-fi∥=0foreachi=1,2,…,m.
Then, for everyf∈C(X), one has stA-limn∥Ln(f)-f∥=0.Proof.
Takeℱ=C(X),X=Y, and F(x,y)=P(x,y). Then, since
(45)Ln(Py;x)-Py(x)=∑i=1m‍gi(y){Ln(fi;x)-fi(x)},
we observe that(46)supx,y∈X|Ln(Py;x)-Py(x)|≤C∑i=1m‍∥Ln(fi)-fi∥,
whereC≔max1≤i≤m∥gi∥. Now, for a given ε>0, consider the following sets:
(47)B≔{n∈ℕ:supx,y∈X|Ln(Py;x)-Py(x)|≥ε},Bi≔{n∈ℕ:∥Ln(fi)-fi∥≥εmC},i=1,2,…,m.
Hence, inequality (46) implies that
(48)B⊂⋃i=1m‍Bi,
which gives, for everyj∈ℕ, that
(49)∑n∈B‍ajn≤∑i=1m‍∑n∈Bi‍ajn.
By (44), we obtain that
(50)stA-limn∑n∈B‍ajn=0,
which gives(51)stA-limn(supx,y∈X|Ln(Py;x)-Py(x)|)=0.
Thus, the last equality means that condition (2) is valid for the function P(x,y). As a result, all hypotheses of Theorem 4 are satisfied.If we takeA=I, the identity matrix, in Corollary 5, then we immediately get the classical result (see, e.g., [3, page 22]).In algebraic case, we consider the following test functions:ei(x)=xi,i=0,1,2. Then we get the next result.Corollary 6 (see Corollary  2 of [8]).
Let{Ln} be a sequence of positive linear operators from C[a,b] into itself. If, for a given nonnegative regular summable matrix A=[ajn],
(52)stA-limn∥Ln(ei)-ei∥=0foreachi=0,1,2,
then, for everyf∈C[a,b], one has stA-limn∥Ln(f)-f∥.Proof.
If we takeX=ℝ, Y=[a,b], ℱ=C[a,b], and F(x,y)=(y-x)2, then we observe that all conditions of Theorem 4 are satisfied.Of course, ifA=C1, the Cesáro matrix, in Corollary 6, then one obtains Theorem  1 of [9]. Furthermore, taking A=I we get the classical theorem (see [2]).Finally, as in [4], Theorem 4 also contains the trigonometric version of Corollary 6 introduced in [10].



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