#### Abstract

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 -statistical approximation for regular summability matrix (see [1, 2]). Let , , be an infinite summability matrix. For a given sequence , the -transform of , denoted by , is given by , provided that the series converges for each . is said to be regular if , whenever . Then , for all . In [3], Dzyubenko and Gilewicz have given the notion.

is nonnegative regular summability matrix. Then is -statistically convergent to , if, for every , .

We denote by the space of all -periodic and continuous functions on . Endowed with the norm , this space is a Banach space, where . Now, recall that, in [4], the th order Ditzian-Totik modulus of smoothness in the uniform metric is given by where is the symmetric th difference. We have to recall the Korovkin type theorem.

Theorem 1 (see [2]). *Let be a sequence of infinite nonnegative real matrices such that and let be a sequence of positive linear operators mapping into . Then, for all , we have uniformly in , if and only if (), uniformly in , where , , and , for all .*

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

Theorem 2. *Let be a sequence of nonnegative regular summability matrices and let be a sequence of positive linear operators mapping into . Then, for all , we have , uniformly in , if and only if (), uniformly in , where , , and , for all .*

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

Let be fixed and sufficiently large. If and , then it is convenient to denote and therefore , for . Recall that is the sign of on .

Now, let us introduce our theorems as follows.

Theorem 3. *Let be a sequence of infinite nonnegative real matrices such that and let be a sequence of positive linear operators mapping into . Then, for all , we have
**
uniformly in , if and only if
**
uniformly in , where , , and , for all . And the constant does not depend on .*

Theorem 4. *Let be a sequence of nonnegative regular summability matrices and let be a sequence of positive linear operators mapping into . Then, if there exists , we have
**
uniformly in , if and only if
**
uniformly in , where , , and , for all .*

#### 2. Proofs of Theorems 3 and 4

*Proof of Theorem 3. *Since () belong to , implications (5) (6) are obvious. Now, assume that (6) holds. Let , and, be a closed subinterval of length of . And let be defined by
and also where and are chosen so that

In [5] Kopotun, we have and , where
is the Lagrange polynomial of degree , which interpolates at , , and . Inequality (11) is an analog of Whitney's inequality for Ditzian-Totik moduli. Using (11) and the above presentations of and , we write, for ,
Taking supremum over and , we obtain
Suppose , let us write sets as follows:
Consequently, we get and implies

*Proof of Theorem 4. *Since () belong to , implications (8) (7) are obvious. Assume that the condition (7) is satisfied. Let and be a closed subinterval of length of ; we have

Now, given , choose , where implied , and define the following set:
Thus,
where polynomial and . Since is -statistically convergent, we can easily show that implies .

Now, let , and using (7) implies
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 -statistical convergence (see [6–8]). If , then there is a sequence of polynomials and -statistically uniformly convergent to on (not uniformly convergent). Observe that Fejer operators may be written in the form of We now consider the linear operator defined by where is a matrix of real numbers and also and are Fourier coefficients. Now, let be a nonnegative regular summability matrice. Assume that the following statements are satisfied: (i);(ii). We get where is the sequence of linear operators given by (21).

In [9], Sakaoğlu and Ünver proved the following theorem by using and denoted the space of all functions defined on , for which , . In this case, the norm of a function in , denoted by , is given by .

Theorem 5 (see [9]). *Let be a nonnegative regular summability matrix and let be an -statistically uniformly bounded sequence of positive linear operators from into and . Then, for any function, if and only if , where , , , and .*

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.

Now, let us introduce our definition as follows.

*Definition 6 (Lebesgue-Stieltjes integral-). *Let be measurable set, be a bounded function, and be nondecreasing function for . For Lebesgue partition of , put and such that measurable function of ; , , and . Also, , , , and , where and . If , where . Then is integral according to for .

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 is positive linear functional and bounded on , is bounded measurable function to . Then, there exists nondecreasing function to such that .*

*Proof. *Assume that functional supremum is as follows:
where converges to ; that is, let be Lebesgue partition such that
where .

Since positive linear functional and bounded on , then
also, respect between sum and Lebesgue-Stieltjes integral- are , we have
as ; hence satisfies Lebesgue-Stieltjes integral- of .

Now, since is functional supremum and satisfies Lebesgue-Stieltjes integral-, and us Definition 6, we have
Note that effect sum on measurable function by using Lebesgue partition
let , choose , and define the following sets:
Then , which gives
we obtain that implies .

Now, in this paper we have proved Riesz's representation theory with Lebesgue-Stieltjes integral-, 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.