#### Abstract

In this article, we purpose to study some approximation properties of the one and two variables of the Bernstein-Schurer-type operators and associated GBS (Generalized Boolean Sum) operators on a symmetrical mobile interval. Firstly, we define the univariate Bernstein-Schurer-type operators and obtain some preliminary results such as moments, central moments, in connection with a modulus of continuity, the degree of convergence, and Korovkin-type approximation theorem. Also, we derive the Voronovskaya-type asymptotic theorem. Further, we construct the bivariate of this newly defined operator, discuss the order of convergence with regard to Peetre’s -functional, and obtain the Voronovskaya-type asymptotic theorem. In addition, we consider the associated GBS-type operators and estimate the order of approximation with the aid of mixed modulus of smoothness. Finally, with the help of the Maple software, we present the comparisons of the convergence of the bivariate Bernstein-Schurer-type and associated GBS operators to certain functions with some graphical illustrations and error estimation tables.

#### 1. Introduction

In [1], Bernstein suggested his polynomials that still inspire many studies today as follows: for any and any .

In 1962, the operators , which are called Bernstein-Schurer, are proposed by Schurer [2] as follows: for any and .

Very recently, many modifications and generalizations of the Bernstein or Bernstein-Schurer operators for univariate and bivariate cases are discussed by many authors. For instance, Acar et al. [3] established local and global approximation results in terms of modulus of continuity for a new type of the Bernstein-Durrmeyer operators on mobile interval. Izgi [4] presented a new type of the Bernstein polynomials and studied several approximation results of the univariate and bivariate of these operators. For the parameter , Chen et al. [5] defined a new generalization of the Bernstein operator and derived the order of convergence and Voronovskaya-type asymptotic relation for the Bernstein operator. Kajla and Acar [6] constructed a new kind of the Bernstein operator and studied a uniform convergence estimate, some direct results involving the asymptotic theorems for these operators. Acar et al. [7] introduced the Kantorovich modifications of the Bernstein operators for bivariate functions using a new integral and obtained the uniform convergence and rate of approximation in terms of modulus of continuity for these operators. Further, for , Cai [8] introduced the Bézier version of the Kantorovich-type Bernstein polynomials and gained the global and direct approximation theorems. Acar and Kajla [9] introduced an extension of the bivariate generalized Bernstein operators with nonnegative real parameters and studied the degree of approximation with regard to Peetre’s -functional and Lipschitz-type functions. Bărbosu [10] demonstrated the uniform convergence and estimated the degree of approximation of the bivariate of the Bernstein-Schurer operators. Căbulea [11] considered the generalizations of the Kantorovich and Durrmeyer type of the Bernstein-Schurer operators and evaluated in connection with the modulus of continuity the order of approximation of these operators. Also, for some recent works, we can refer the readers to ([12–23]).

By the motivation of the all the above-mentioned works, we define the univariate Bernstein-Schurer-type operators on a symmetrical mobile interval. Let the intervals be , , and be the set of all continuous and bounded functions on For a function and the univariate Bernstein-Schurer operators are defined as where .

The goal of the present work is to obtain some approximation features of the operators given by (3). We show the uniform convergence, estimate the degree of convergence with the help of modulus of continuity, and prove the Voronovskaya-type asymptotic theorem for the (3) operators. Next, we define the bivariate of (3) operators, compute the order of convergence by using Peetre’s -functional, and derive the Voronovskaya-type asymptotic theorem for the bivariate case. Further, we construct the associated GBS type of bivariate operators and estimate their degree of convergence in terms of mixed modulus of smoothness. Finally, by the help of the Maple software, we give comparisons of the convergence of bivariate of (3) operators and related GBS operators to the certain functions with some graphics and error estimation tables.

#### 2. Main Results

Lemma 1. *Let the operators be defined by (3). Then, for all , the following moments verify
*

*Proof. *From (3), it becomes
The last two identities can be obtained by applying similar methods; hence, we have omitted the details.

Lemma 2. *For all , we obtain the following central moments:
*

*Proof. *The proof of this lemma can be directly obtained by using the linearity of (3) operators and as a consequence of Lemma 1.

Corollary 3. *For all , the following identities hold:
**In the next theorem we show the uniform convergence of (3) operators. As it is known, the space denotes the real-valued continuous functions on , and it is equipped with the norm for a function as follows:
*

Theorem 4. *Let the operators be given by (3). Then, for all , converges to uniformly on .*

*Proof. *From (4), it is obvious that
By (5), we arrive
Similarly, using (6), then
Hence, according to the Korovkin theorem [24], the (3) operators converge uniformly to on

Further, we will obtain the degree of approximation of (3) operators. Let the modulus of continuity for a function be given by

Since has some useful properties which can be found in [25].

Theorem 5. *Let Then, for every , the following inequality is verified:
where *

*Proof. *Taking into account the following common property of the modulus of continuity:
by the linearity of the operator (3), then
Utilizing the Cauchy-Schwarz inequality yields
If we take thus Theorem 5 is proven.

Theorem 6. *Let the operators be given by (3). Then, for any such that , the following identity holds:
uniformly on *

*Proof. *Suppose that and are fixed. By the Taylor formula, hence
where is a form of Peano of the rest term, and since

Operating to (22), then
From Lemma 2, it becomes
Applying the Cauchy-Schwarz inequality, one has
Owing to thus
uniformly on with Theorem 4.

Combining (25) and (26) and by Lemma 2, we get
Hence,
which gives the proof.

#### 3. Construction of the Bivariate Bernstein-Schurer-Type Operators

Let the intervals be , and by , we denote the set of all real-valued continuous functions on , and it is equipped with the norm

We define the bivariate of operators given by (3) as follows: where , and

It can be seen that the operators given by (29) are positive and linear.

Lemma 7. *Let with , be the bivariate test functions. Then, one has
*

*Proof. *The proof of the above equalities can be reached easily as a consequence of Lemma 1 and by (29); hence, we have omitted the details.

Corollary 8. *In view of Lemma 7, the following relations hold true:
*

Theorem 9. *Let the operators be given by (29). Then for any , we arrive at
*

*Proof. *It is seen from the following that
as Thus, these results complete the proof, as required by the Volkov theorem [26].

Moreover, for the operators given by (29), we want to derive the Voronovskaya-type asymptotic theorem and estimate the degree of convergence with the help of Peetre’s -functional.

Suppose that represents the space of all functions of such that . The norm on and Peetre’s -functional of are given as follows, respectively. where .

For a constant the following inequality holds, where denotes the second order of the modulus of continuity of Also, for the ordinary modulus of continuity is defined as

Theorem 10. *Suppose that . Then, the following relation holds
uniformly on *

*Proof. *For the arbitrary using the Taylor formula, it becomes
for where and as

Operating on (38) yields
If we use the Cauchy-Schwarz inequality to the last part of (39), one has
Considering Theorem 9 and because of as then
uniformly on Also, from Corollary 8, it is clear to see
Further, by Corollary 3, it follows that
Thus,
Hence, the desired sequel is arrived as follows:
uniformly on .

Theorem 11. *Suppose that . Then, the following inequality is verified:
where is a constant and independent of and and , .*

*Proof. *Let us define the following auxiliary operators:
By Lemma 7, we obtain
Suppose that and ; by the Taylor formula, we may write
Now, operating on (49), we get