We present a Korovkin type approximation theorem for a sequence of positive linear operators defined on the space of all real valued continuous and periodic functions via A-statistical approximation, for the rate of the third order Ditzian-Totik modulus of smoothness. Finally, we obtain an interleave between Riesz's representation theory and Lebesgue-Stieltjes integral-i, for Riesz's functional supremum formula via statistical limit.

1. Introduction and Main Results

Some will accept the notes and definitions used in this paper. The concept of A-statistical approximation for regular summability matrix (see [1, 2]). Let A=(ank), n,k=1,2,…, be an infinite summability matrix. For a given sequence x=(xk), the A-transform of x, denoted by Ax=(Ax)n, is given by (Ax)n=∑k=1∞ankxk, provided that the series converges for each n. A is said to be regular if limn→∞(Ax)n=L, whenever limx=L. Then limn→∞ank=0, for all k∈N. In [3], Dzyubenko and Gilewicz have given the notion.

A is nonnegative regular summability matrix. Then x is A-statistically convergent to L, if, for every ∈>0, limn→∞∑k:|xk-L|≥∈ank=0.

We denote by C2π(ℛ) the space of all 2π-periodic and continuous functions on ℛ. Endowed with the norm ∥·∥2π, this space is a Banach space, where ∥f∥2π=sup{|f(t)|:f∈C2π(ℛ),t∈ℛ}. Now, recall that, in [4], the mth order Ditzian-Totik modulus of smoothness in the uniform metric is given by
(1)ωmϕ(f,δ,[a,b])=sup0<h≤δ∥Δhϕ(x)m(f,x,[a,b])∥[a,b],
where
(2)Δhm(f,x,[a,b])={∑i=0m(mi)(-1)m-i×f(x-mh2+ih),ifx±mh2∈[a,b],0,o.w,
is the symmetric mth difference. We have to recall the Korovkin type theorem.

Theorem 1 (see [<xref ref-type="bibr" rid="B2">2</xref>]).

Let A=(An)n≥1 be a sequence of infinite nonnegative real matrices such that supn,k∑j=1∞akjn<∞ and let {Lj} be a sequence of positive linear operators mapping C2π(ℛ) into C2π(ℛ). Then, for all f∈C2π(ℛ), we have limk→∞∑j=1∞akjn∥Ljf-f∥2π=0 uniformly in n, if and only if limk→∞∑j=1∞akjn∥Ljfi-fi∥2π=0 (i=1,2,3), uniformly in n, where f1(t)=1, f2(t)=cost, and f3(t)=sint, for all t∈ℛ.

It is worth noting that the statistical analog of Theorem 1 has been studied by Radu [2], as follows.

Theorem 2.

Let A=(An)n∈N be a sequence of nonnegative regular summability matrices and let {Lj} be a sequence of positive linear operators mapping C2π(ℛ) into C2π(ℛ). Then, for all f∈C2π(ℛ), we have stA-limj→∞∥Ljf-f∥2π=0, uniformly in n, if and only if stA-limj→∞∥Ljfi-fi∥2π=0 (i=1,2,3), uniformly in n, where f1(t)=1, f2(t)=cost, and f3(t)=sint, for all t∈ℛ.

The following notations are used this paper (see [5, 6]).

Let n be fixed and sufficiently large. If yi∈Ij(i) and 1≤i≤k, then it is convenient to denote
(3)yi′=xj(i)+1,yi′′=xj(i)-2,Ii′=[yi′,yi′′]=Ij(i)+1∪Ij(i)∪Ij(i)-1=[xj(i)+1,xj(i)]∪[xj(i),xj(i)-1]∪[xj(i)-1,xj(i)-2],ρi=[yi+yi′2,yi+yi′′2],for1≤i≤k,53hj(i)=53(xj(i)-1-xj(i))<(xj(i)-2-xj(i)+1)=|Ii′|=2|ρi|=(yi′′-yi′)=(xj(i)-2-xj(i)+1)<7,hj(i)=7(xj(i)-1-xj(i)),1≤i≤k,
and therefore |Ii′|~|ρi|~hj(i), for x∈Ii′. Recall that
(4)sgn(f(x))={1;ifx∈[a,b],-1;ifx∉[a,b],
is the sign of f on [a,b].

Now, let us introduce our theorems as follows.

Theorem 3.

Let A=(An)n≥1 be a sequence of infinite nonnegative real matrices such that supn,k∑j=1∞akjn<∞ and let {Lj} be a sequence of positive linear operators mapping C2π(ℛ) into C2π(ℛ). Then, for all f∈C2π(ℛ), we have
(5)∑j=1∞akjn∥Ljf-f∥2π≤cω3ϕ(f,πn,[-π,π]),
uniformly in n, if and only if
(6)∑j=1∞akjn∥Ljfζ-fζ∥2π≤cω3ϕ(fζ,πn,[-π,π]),ζ=1,2,3,
uniformly in n, where f1(t)=1, f2(t)=(t-yi′)/(yi′′-yi), and f3(t)=(t-yi′′)/(yi-yi′), for all t∈ℛ. And c the constant does not depend on j.

Theorem 4.

Let A=(An)n∈N be a sequence of nonnegative regular summability matrices and let {Lj} be a sequence of positive linear operators mapping C2π(ℛ) into C2π(ℛ). Then, if there exists f∈C2π(ℛ), we have
(7)stA-limn∥Ljf-f∥2π≥c(j)ω3ϕ(f,πn,[-π,π]),
uniformly in n, if and only if
(8)stA-limn∥Ljfζ-fζ∥2π≥c(j)ω3ϕ(fζ,πn,[-π,π])llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllζ=1,2,3,
uniformly in n, where f1(t)=1, f2(t)=(t-yi′)/(yi′′-yi), and f3(t)=(t-yi′′)/(yi-yi′), for all t∈ℛ.

2. Proofs of Theorems <xref ref-type="statement" rid="thm1.3">3</xref> and <xref ref-type="statement" rid="thm1.6">4</xref>Proof of Theorem <xref ref-type="statement" rid="thm1.3">3</xref>.

Since fζ (ζ=1,2,3) belong to C2π(ℛ), implications (5) ⇒ (6) are obvious. Now, assume that (6) holds. Let f∈C2π(ℛ), and, I be a closed subinterval of length 2π of ℛ. And let Lj be defined by
(9)Lj(x)=xn-yiyi′′-yi′(x-yi′yi′′-yiLj(yi′′)-x-yi′′yi-yi′Lj(yi′)),yi∈[yi+yi′2,yi+yi′′2]fori=1,2,3,
and also where Lj(yi′) and Lj(yi′′) are chosen so that(10)Lj(yi′)={Cω3ϕ(f,n-1)sgn(f(yi′));if|f(yi′)|≤cω3ϕ(f,n-1),f(yi′);o.w,Lj(yi′′)={Cω3ϕ(f,n-1)sgn(f(yi′′));if|f(yi′′)|≤cω3ϕ(f,n-1),f(yi′′);o.w.

In [5] Kopotun, we have |f(x)-L(f;x)|≤cω3ϕ(f,n-1) and x∈I, where
(11)L(f;x)=L(f;x∣yi,yi′,yi′′)=xn-yiyi′′-yi′(x-yi′yi′′-yiLj(yi′′)-x-yi′′yi-yi′Lj(yi′))
is the Lagrange polynomial of degree ≤2, which interpolates f at yi, yi′, and yi′′. Inequality (11) is an analog of Whitney's inequality for Ditzian-Totik moduli. Using (11) and the above presentations of Lj and L(f;x), we write, for x∈I,
(12)|Lj(f;x)-f(x)|≤|Lj(f;x)-L(f;x)|+|L(f;x)-f(x)|≤|(xn-yi)(x-yi′)(yi′′-yi′)(yi′′-yi)||Lj(yi′′)-f(yi′′)|+|(xn-yi)(x-yi′′)(yi′′-yi′)(yi′′-yi)|×|Lj(yi′)-f(yi′)|+|L(f;x)-f(x)|.
Taking supremum over x and =1/(ω3ϕ(f,n-1)sgn(f)), we obtain
(13)∥Lj(f;x)-f(x)∥2π≤∥Lj(f;x)-L(f;x)∥2π+∥L(f;x)-f(x)∥2π≤|(xn-yi)(x-yi′)(yi′′-yi′)(yi′′-yi)|∥Lj(yi′′)-f(yi′′)∥2π+|(xn-yi)(x-yi′′)(yi′′-yi′)(yi′′-yi)|∥Lj(yi′)-f(yi′)∥2π+∥L(f;x)-f(x)∥2π≤c(K,yi′′)+c(K,yi′)+cω3ϕ(f,n-1,[-1,1]).
Suppose B>0, let us write sets as follows:
(14)ϑ={∥Lj(t-yi′′yi-yi′;x)-x-yi′′yi-yi′∥2πj:∥Lj(1;x)-1∥2π+∥Lj(t-yi′yi′′-yi;x)-x-yi′yi′′-yi∥2π+∥Lj(t-yi′′yi-yi′;x)-x-yi′′yi-yi′∥2π≥KB},ϑ1={j:∥Lj(1;x)-1∥2π≥KB},ϑ2={j:∥Lj(t-yi′yi′′-yi;x)-x-yi′yi′′-yi∥2π≥KB},ϑ3={j:∥Lj(t-yi′′yi-yi′;x)-x-yi′′yi-yi′∥2π≥KB}.
Consequently, we get ϑ⊂ϑ1∪ϑ2∪ϑ3 and ∑j∈ϑakjn≥∑j∈ϑ1akjn≥∑j∈ϑ2akjn≥∑j∈ϑ3akjn implies
(15)∑j=1∞akjn∥Ljf-f∥2π≤cω3ϕ(f,πn,[-π,π]).

Proof of Theorem <xref ref-type="statement" rid="thm1.6">4</xref>.

Since fζ (ζ=1,2,3) belong to C2π(ℛ), implications (8) ⇒ (7) are obvious. Assume that the condition (7) is satisfied. Let f∈C2π(ℛ) and I be a closed subinterval of length 2π of ℛ; we have
(16)stA-limn∥Ljf-f∥2π≥c(j)ω3ϕ(f,πn,[-π,π]).

Now, given K(j)>0, choose B>0, where B=sup{|f(x)|:x∈I} implied K<B, and define the following set:
(17)ϑ={+∥Lj(t-yi′′yi-yi′;x)-x-yi′′yi-yi′∥2πj:∥Lj(1;x)-1∥2π+∥Lj(t-yi′yi′′-yi;x)-x-yi′yi′′-yi∥2π+∥Lj(t-yi′′yi-yi′;x)-x-yi′′yi-yi′∥2π≥KB}.
Thus,
(18)stA-limn∥Ljp3(t)-p3(x)∥2π=stA-limn∥Lj(t-yi′′yi-yi′;x)-x-yi′′yi-yi′∥2π≥(K-K∘)B`,
where p3(x)=(x-yi′′)/(yi-yi′)∈C2π(ℛ) polynomial and x∈ℛ. Since x is A-statistically convergent, we can easily show that ϑ⊃ϑ1⊃ϑ2⊃ϑ3 implies ∑j∈ϑakjn≥∑j∈ϑ1akjn≥∑j∈ϑ2akjn≥∑j∈ϑ3akjn.

Now, let B`=ω3ϕ(fζ,π/n,[-π,π]), and using (7) implies
(19)stA-limn∥Ljp3(t)-p3(x)∥2π≥c(j)ω3ϕ(p3,πn,[-π,π]).
This is a complete proof.

3. Application to Functional Approximation

In this section we give some applications which satisfy our theorems, but it's not the classical Korovkin theorem. It has been treated with the Weierstrass second approximation theorem via A-statistical convergence (see [6–8]). If f∈C2π(ℛ), then there is a sequence of polynomials and A-statistically uniformly convergent to f on [-π,π] (not uniformly convergent). Observe that Fejer operators may be written in the form of
(20)Fn(f;x)=a∘2+∑k=1nn-kn(akkx-yi′yi′′-yi+bkkx-yi′′yi-yi′).
We now consider the linear operator Tn defined by
(21)Tn(f;x)=a∘2+∑k=1nξk(n)(akkx-yi′yi′′-yi+bkkx-yi′′yi-yi′),
where {ξk(n)}(n=1,2,…;k=1,2,…,n) is a matrix of real numbers and also ak and bk are Fourier coefficients. Now, let A=(ank) be a nonnegative regular summability matrice. Assume that the following statements are satisfied:

stA-limnξ1(n)=1;

(1/2)+∑k=1nξk(n)(t-yi′)/(yi′′-yi)≥c(n)ω3ϕ(f,π/n,[-π,π]). We get
(22)stA-limn∥Tn(f;x)-f(x)∥2π≥c(n)ω3ϕ(f,πn,[-π,π]),∀f∈C2π(ℛ),
where {Tn} is the sequence of linear operators given by (21).

In [9], Sakaoğlu and Ünver proved the following theorem by using LP[a,b;c,d] and denoted the space of all functions f defined on [a,b]×[c,d], for which ∫cd∫ab|f(x,y)|Pdxdy<∞, 1≤P<∞. In this case, the LP norm of a function f in LP[a,b;c,d], denoted by ∥f∥P, is given by ∥f∥P=(∫cd∫ab|f(x,y)|Pdxdy)1/P.

Theorem 5 (see [<xref ref-type="bibr" rid="B9">9</xref>]).

Let A=(ajn) be a nonnegative regular summability matrix and let {Tn} be an A-statistically uniformly bounded sequence of positive linear operators from LP[a,b;c,d] into LP[a,b;c,d] and 1≤P<∞. Then, for any functionf∈LP[a,b;c,d], stA-limn∥Tn(f;x,y)-f(x,y)∥P=0 if and only if stA-limn∥Tn(fi;x,y)-fi(x,y)∥P=0, i=1,2,3,4 where f1(t,v)=1, f2(t,v)=t, f3(t,v)=v, and f4(t,v)=t2+v2.

The theory of the Lebesgue integral can be developed in several distinct ways (see [10, 11]). Only one of these methods will be discussed here.

Let S be measurable set, f:S→R be a bounded function, and gi:S→R be nondecreasing function for i∈I. For 𝒫 Lebesgue partition of S, put LS_(f,𝒫,g_)=∑j=1n∏i∈Imjgi(μ(Sj)) and LS¯(f,𝒫,g_)=∑j=1n∏i∈IMjgi(μ(Sj)) such that μ measurable function of S; mj=inf{f(x):x∈Sj}, Mj=sup{f(x):x∈Sj}, and g_=g1,g2,…. Also, gi(xj)-gi(xj-1)>0, LS_(f,𝒫,g_)≤LS¯(f,𝒫,g_), ∏i∈I∫i_fdg_=sup{LS_(f,g_)}, and ∏i∈I∫i¯fdg_=inf{LS¯(f,g_)}, where LS_(f,g_)={LS_(f,𝒫,g_):𝒫partofsetS} and LS¯(f,g_)={LS¯(f,𝒫,g_):𝒫partofsetS}. If ∏i∈I∫i_fdg_=∏i∈I∫i¯fdg_, where dg_=dg1×dg2×…×dgn…. Then f is integral ∫i according to gi for i∈I.

Now, we can provide our theorem as follows as a case which is an illustrative application of approximation theory in functional analysis using functional supremum to limit convergence that acts as support and reinforcement of the concept of Riesz's representation.

Theorem 7.

If a sequence Gn(f) is positive linear functional and bounded on C(S), f is bounded measurable function to S. Then, there exists nondecreasing function to S such that stA-limμ(S)→0(supnGn(f)-f)=0.

Proof.

Assume that functional supremum Gn is as follows:
(23)supnGn(f)=supn∏i∈I∫ifdφt,n_,
where φt,n(x)=(1-n(x-t))/(yi′′-yi′) converges to r∈R; that is, let 𝒫={Sℓ}ℓ=0m be Lebesgue partition such that
(24)sup{μ(Sℓ):ℓ=0,…,m}<12δ(ε),12<inf{μ(Sℓ):ℓ=0,…,m}φit,n(μ(Sℓ))≤G(πt,n(μ(Sℓ)))≤φit,n(μ(Sℓ))+εm∥f∥,
where πt,n(x)=φt,n(x)(yi′′-yi′).

Since G positive linear functional and bounded on C(S), then
(25)|G(f)-G(f(t1)πt1,n+∑ℒ=2mf(tℒ)πtℒ,n(μ(Sℓ)))|<εm∥f∥,
also, respect between sum LS(f,𝒫,φt,n_) and Lebesgue-Stieltjes integral-i are 2ε, we have
(26)|∏i∈I∫ifdφt,n_-G(f)|≤|∏i∈I∫ifdφt,n_-G(f(t1)πt1,n+∑ℒ=2mf(tℒ)πtℒ,n(μ(Sℓ)))|+|G(f)-G∑ℒ=2mf(tℒ)(f(t1)πt1,n+∑ℒ=2mf(tℒ)πtℒ,n(μ(Sℓ)))|<ε`m
as m→∞; hence G satisfies Lebesgue-Stieltjes integral-i of f.

Now, since Gn(f) is functional supremum and satisfies Lebesgue-Stieltjes integral-i, and us Definition 6, we have
(27)supnGn(f)-f=supn∏i∈I∫ifdφt,n_-f=supn(sup∑j=1n∏i∈Imjφit,n(μ(Sj)))-f=supn(sup∑j=1n∏i∈Iinfjf(x)φit,n(μ(Sj)))-f=supn(inf∑j=1n∏i∈Isupjf(x)φit,n(μ(Sj)))-f=supn(inf∑j=1nsupjf(x)∏i∈Iφit,n(μ(Sj)))-f=supn(inf∑j=1nsupjf(x)×([(1-n(μ(S1)-t)yi′′-yi′)·(1-n(μ(S2)-t)yi′′-yi′)⋯]inf∑j=1nsupjf(x))-f=supn(inf(×[(1-n(μ(S1)-t)yi′′-yi′)sup1f(x)×[(1-n(μ(S1)-t)yi′′-yi′)·(1-n(μ(S2)-t)yi′′-yi′)⋯]+sup2f(x)[(1-n(μ(S1)-t)yi′′-yi′)·(1-n(μ(S2)-t)yi′′-yi′)⋯]+⋯+supnf(x)×[(1-n(μ(S1)-t)yi′′-yi′)·(1-n(μ(S2)-t)yi′′-yi′)⋯]))-f.
Note that effect sum on measurable function μ(Sj) by using Lebesgue partition
(28)limμ(S)→0∏i∈I∫ifdφt,n_=f;
let n∈N, choose 𝒦n>0, and define the following sets:
(29)𝕃={μ(S):(supnGn(f)-f)≥𝒦n},𝕃1={μ(S):(supnGn(f1)-f1)≥𝒦n3},𝕃2={μ(S):(supnGn(f2)-f2)≥𝒦n3},𝕃3={μ(S):(supnGn(f3)-f3)≥𝒦n3};
Then 𝕃⊂𝕃1∪𝕃2∪𝕃3, which gives
(30)∑μ(S)⊂𝕃akjn≤∑μ(S)⊂𝕃1akjn∪∑μ(S)⊂𝕃2akjn∪∑μ(S)⊂𝕃3akjn;
we obtain that stA-limμ(S)→0∑μ(S)⊂𝕃akjn=0 implies stA-limμ(S)→0(supnGn(f)-f)=0.

Now, in this paper we have proved Riesz's representation theory with Lebesgue-Stieltjes integral-i, by using Korovkin type approximation which is one of the threads in the development of Riesz's theorem to support the definition of Lebesgue integral, Rudin [10]. This integration toxicity ratio for the world on behalf of the French Lebesgue, who came in his thesis for a doctorate in 1902.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author is grateful for hospitality at the University of Kufa. He thanks his fellows for the fruitful discussions while preparing this paper. He was partially supported by University of Al-Muthanna.

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