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 , where denotes the distance between and .
Proof. Let be the closure of . Using the properties of infimum, there is at least a point such that . By the triangle inequality we have Choosing and and using the well-known Hölder inequality, we have This completes the proof. ☐
7. Numerical Examples
In this section, we will analyze the theoretical results presented in the previous sections by numerical examples.
Let , , and . The convergence of the operators to function is shown in Figure 1. The error of approximation is given in Figure 2. Meantime, we compute the error of approximation while at points in Table 1.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant Nos. 11626031, 11861025, and 11501143), the Key Natural Science Research Project in Universities of Anhui Province (Grant No. KJ2019A0572 and Grant No. KJ2019A0580), the Philosophy and Social Sciences General Planning Project of Anhui Province of China (Grant No. AHSKYG2017D153), the Natural Science Foundation of Anhui Province of China (Grant No. 1908085QA29), and the Doctoral research project of Guizhou Normal University (No. GZNUD[2017]27).