#### Abstract

In this work, we extend the works of F. Usta and construct new modified -Bernstein operators using the second central moment of the -Bernstein operators defined by G. M. Phillips. The moments and central moment computation formulas and their quantitative properties are discussed. Also, the Korovkin-type approximation theorem of these operators and the Voronovskaja-type asymptotic formula are investigated. Then, two local approximation theorems using Peetre’s -functional and Steklov mean and in terms of modulus of smoothness are obtained. Finally, the rate of convergence by means of modulus of continuity and three different Lipschitz classes for these operators are studied, and some graphs and numerical examples are shown by using Matlab algorithms.

#### 1. Introduction

In [1], Phillips introduced -analogue of Bernstein operators as follows: where , , and . Later, generalizations of -Bernstein operators (1) attracted a lot of interest and were constructed and researched widely by a number of researchers. For instance, in [2], Mahmudov and Sabancigil introduced -Bernstein-Kantorovich operators and studied local and global approximation properties. In [3], Acu et al. defined modified -Bernstein-Kantorovich operators and established the shape-preserving properties of these operators, e.g., monotonicity and convexity. Some other papers also mention Bernstein operators with parameter(s) and their modification: -Bernstein operators [4], -Bernstein operators [5–8], -Bernstein operators [9], -Bernstein operators [10], -Bernstein-Stancu operators [11], -Bernstein-Kantorovich operators [12], generalized Bernstein operators [13], and so on.

In this article, we consider -analogue of the following new Bernstein operators constructed by the second central moment of the classic Bernstein operators which was given in [14].

There are many papers about the research and application of -operators, and we mention some of them: -Bleimann-Butzer-Hahn operators [15], Bivariater -Meyer-König-Zeller operators [16], -Baskakov operators [17, 18], -Meyer-König-Zeller-Durrmeyer operators [19], -Phillips operators [20, 21], -Szász operators [22], -Bernstein operators [23], and so on. All this achievement motivates us to construct the -analogue of the operators (2). Before continuing further, let us recall some useful concepts and notations from -calculus, which can be found in [24]. For nonnegative integer , the -integer , -factorial , and -binomial coefficients are defined by

Further, -power basis can be defined by

The -derivative of a function can be defined by and provided exists. High-order -derivatives can be defined by , , . The formula for the -derivative of a product is

The -analogue of new-Bernstein operators (2) on is defined by the following: where , , , and .

The rest of the paper is organized as follows: In Section 2, we get the basic results by the moment computation formulas. And the first, second, fourth, and sixth order central moment computation formulas and limit equalities are also computed. In Sections 3 and 4, we investigate the Korovkin-type approximation theorem and the Voronovskaja-type asymptotic formula for the operators (7). In Section 5, we obtain two local approximation theorems using Peetre’s -functional and Steklov mean and in terms of modulus of smoothness. In Section 6, we study the rate of convergence by means of modulus of continuity and three different Lipschitz classes for these operators (7). In Section 7, we show some graphs and numerical examples to analyze the theoretical results by using Matlab algorithms.

#### 2. Auxiliary Lemmas

In this section, we present certain auxiliary results which will be used to prove our main theorems for the operators (7). Using the lemma in ([25], Lemma 2), we have the moment computation formulas for the operators (1):

Lemma 1. *If we define , then there holds the following relation:
*

Now, we give the moment relation between the operators (1) and the operators (7) as follows:

Lemma 2. *If we define , then there holds the following relation:
where , , , and .*

*Proof. *By the definition of and , we can obtain
Next, by (9) and (10), we can obtain
Thus, we complete the proof of (11). Finally, by (6), (9), and (11), we can obtain
We complete the proof of the Lemma 2. ☐

Then, the following lemma can be obtain immediately:

Lemma 3. *For , , and , we have
*

Lemma 4. *The sequence satisfies , such that and as ; then, for any , we have
*

*Proof. *First, we prove the limit . In fact, for any , such sufficiently large that . But for such that , we easily have . Applying Lemma 3, we can directly obtain (17) and (18). As , using Lemma 1, we can rewrite
Applying (12), we can obtain
Combining Lemma 3 and , we can obtain
where , , , , , , , and . Hence, , , , , , , , and as , we have
Combining , we can obtain (19). ☐

Lemma 5. *For , , and , we can have , where denotes the set of all real-valued bounded and continuous functions defined on , endowed with the norm .*

*Proof. *In view of (7) and Lemma 3, for any , we have
Taking supremum over all , we obtain the required result. ☐

#### 3. Korovkin Approximation Theorem

Theorem 6. *Let the sequence satisfy , for any ; then, the sequence converges uniformly to on if and only if as .*

*Proof. *Let and as ; then, we have as . By the Korovkin theorem ([26], p. 8, Theorem 6), it is sufficient to prove the three following limit equalities:
We can obtain these three limit equalities easily by Lemma 3. Thus, we get that the sequence converges uniformly to on .

We prove the converse result by contradiction. Assume that does not tend to as ; then, it must contain a subsequence with , such that . Thus,
Taking and in and using Lemma 3, we can obtain
as . This leads to a contradiction; hence, as . The proof is completed. ☐

#### 4. Voronovskaja-Type Theorem

In this section, we give a Voronovskaja-type asymptotic formula for the operators (7) by means of the first, second, and fourth central moments.

Theorem 7. *Under the condition of Lemma 4 and . Suppose that exists at a point ; then, we have
*

*Proof. *Applying the Taylor’s expansion formula for , we have
where
Using the L’Hospital’s Rule,
Thus, . Applying to both sides of (29) and using Lemma 3, we have
Applying the Cauchy-Schwarz inequality, we have
By Theorem 6, we can obtain
From (19), (33), and (34), we have
Combining (17), (18), and (35), we complete the proof of Theorem 7. ☐

Corollary 8. *Under the condition of Lemma 4 and , then
uniformly in .*

#### 5. Local Approximation

Firstly, we recall the first and second order modulus of continuity of are defined, respectively, by

The Peetre’s -functional is defined by

By ([26], p. 177, Theorem 2.4), there exists an absolute constant depending only on such that

Theorem 9. *Under the condition of Lemma 4, then for all and , there exists an absolute positive constant such that
*

*Proof. *For , we define new operators by
By Lemma 3, we can obtain immediately
For any and , by the Taylor’s expansion formula, we can obtain
Applying the operators (7) to both sides of the above equation, we have
Hence,
By Lemma 5, we have
For any and , combining (45) and (46), we obtain
Taking infinum on the right hand side over all , using (39), we obtain the desired assertion. ☐

Now, we discuss local approximation theorems for the operators (7) by Steklov mean. For any and , the Steklov mean is defined by

In direct computation, it is proved that while .

Theorem 10. *Under the condition of Lemma 4, then for all and ,
*

*Proof. *For , by the definition of the Steklov mean, we can write
By Lemma 5 and (49), we have
Now, by the Taylor’s expansion formula for , we have
Combining (49) and (50), we have
Hence,
for . Setting , we obtain the desired result. ☐

#### 6. Rate of Convergence

First, we discuss the rate of convergence of the operators (7) by means of the modulus of continuity .

Theorem 11. *Under the condition of Lemma 4, then for all and ,
*

*Proof. *Using ([26], p. 41, (6.5)), for all and , we have
Applying the monotonicity and the linearity of and Cauchy-Schwarz inequality, for any , we have
by taking . We complete the proof of Theorem 11. ☐

Theorem 12. *Under the condition of Lemma 4, then for all and ,
*

*Proof. *Applying to both sides of , we have
with the help of Cauchy-Schwartz inequality and mean value theorem. Taking and by Lemma 3, we can get the desired result. ☐

Next, we discuss the rate of convergence of the operators (7) by means of three Lipschitz classes: , , and . A function belongs to , if the condition is satisfied, where is a positive constant depending only on and .

Theorem 13. *Under the condition of Lemma 4, then for all and ,
*

*Proof. *According to the monotonicity and the linearity of the operators (7) and taking into account that , we can obtain
Applying well-known Hölder’s inequality with and , we can get
We obtain the required result. ☐

In [27], Özarslan and Aktulu constructed the following Lipschitz-type space with two distinct parameters as follows: where and is a positive constant depending only on , and .

Theorem 14. *Under the condition of Lemma 4, then for all and ,
*

*Proof. *Applying the well-known Hölder inequality with and , we have
Thus, the proof of Theorem 14 is completed. ☐

A function belongs to , if the condition is satisfied, where is a positive constant depending only on and .

Theorem 15. *Under the condition of Lemma 4, then for all and ,
*