Abstract
In the present paper, Durrmeyer type -Bernstein operators via (, )-calculus are constructed, the first and second moments and central moments of these operators are estimated, a Korovkin type approximation theorem is established, and the estimates on the rate of convergence by using the modulus of continuity of second order and Steklov mean are studied, a convergence theorem for the Lipschitz continuous functions is also obtained. Finally, some numerical examples are given to show that these operators we defined converge faster in some cases than Durrmeyer type (, )-Bernstein operators.
1. Introduction
In 2016, Mursaleen et al. [1] proposed the following (, )-analogue of Bernstein operators: where are (, )-Bernstein basis functions and defined as
They also introduced and studied some important approximation properties of the Stancu type of operators (1) in [2]. After their construction, there are more and more papers on the study of (, )-analogue of Bernstein type operators, we mention some of them as [3–11], we also mention some other positive linear operators as [12–19].
Very recently, Cai et al. [20] proposed the following -Bernstein operators based on (, )-integers as where
are defined in (2), , , , and . They also constructed the (, )-analogue of Kantorovich type -Bernstein operators and investigated their -statistical convergence properties in [21].
Inspired by above research, based on (3), we introduce Durrmeyer type -Bernstein operators via (, )-calculus as where , , are defined in (4) and are defined in (2). Apparently, when , operators become to which are defined as follows in [11], where . We will show that operators converge to faster than operators in some cases of by some numerical examples in Section 4, say, we have more modeling flexibility when adding the parameter .
We first mention some definitions based on (, )-integers; details can be found in [22–26]. For any fixed real number and , the (, )-integers are defined by (, )-factorial and (, )-binomial coefficients are defined as follows:
The (, )-power basis and are defined by
Let , then (, )-integration of a funciton is defined by
This paper is mainly organized as follows: in Section 2, we estimate some moments and central moments of in order to obtain our main results; in Section 3, we study a Korovkin type approximation theorem and estimate the rate of convergence of to by using the second order modulus of smoothness, Peetre’s -functional, Steklov mean function, and Lipschitz class function; in Section 4, we give some numerical experiments to verify our theoretical results; in the final section, a conclusion is given.
2. Some Lemmas
Before giving our main results, we need the following lemmas.
Lemma 1. Let , we have
Proof. According to the following equations of Lemma 1 in [11] and the fact that , we get the proof of Lemma 1 easily.
Lemma 2. Let , , , and , then for the operators , we have
Proof. By (5), (11), and Lemma 2 of [20], we have Next, by the fact that , (5) and (12), we get Then, the desired of (16) can be obtained by Lemma 2 and Lemma 3 of [20] and easy computations. Finally, by (5) and (13), we have Using , we obtain We can get (17) by Lemma 2–4 of [20] and some computations. Lemma 2 is proved.
Lemma 3. Let , , , and , then we have
Proof. We can obtain (22) easily by (15) and (16). For , we have For , we have On one hand, since we get On the other hand, we have with the fact that . Combing (25)–(29), we have Thus, the desired result of (23) is proved. Finally, by Lemma 2 and the linear property of , we have where is some function related to , and , and we will estimate it in two cases. For , we have For , we have From the above two equations (32) and (33), we obtain Combing (31), (33), and (34), we get Thus, we arrive at (24). Lemma 3 is proved.
Lemma 4. (See [6]).
Let sequences , such that , , as. The following statements are true
(A)If and then .(B)If and then .(C)If and then .
3. Rate of Convergence
In the sequel, let sequences and satisfy the conditions of Lemma 4. We first give a Korovkin type approximation theorem for .
Theorem 5. Let be a continuous function on , and , then converge uniformly to on.
Proof. Since the hypothesis of sequences and , we know that as . It is easy to get combining the relation . Therefore, we obtain the desired result due to the well-known Korovkin theorem (see [27], pp. 8-9).
Let be a continuous function on and endowed with the norm . Peetre’s -functional is defined by
where and . The second order modulus of smoothness is defined as
We know that there is a relationship between and , that is
where is a positive constant. The modulus of continuity is denoted by
Then, the rate of convergence of to is given as follows.
Theorem 6. Let be a continuous function on , , and , we have where is a positive constant, and are defined in (23) and (24).
Proof. Let us define auxiliary operators which preserve linear functions as where is defined in (22). Obviously, Set , by Taylor’s expansion, we have Applying to (43) and by (42), we obtain Thus, by (41), we have where and are defined in (23) and (24). According to (41), (5), and (15), we have
Using (41), (45), and (46), we get
Taking the infimum on the right hand side over all , we have
Therefore, we obtain where is a positive constant, and are defined in (23) and (24). Theorem 6 is proved.
Let be a continuous function on , the Steklov mean function is defined as
One can write by the fact that is continuous on . It is obvious that where . If is continuous on , so are and where . Details can be found in [28].
Now, we apply Steklov mean to prove the following theorem.
Theorem 7. Let be a continuous function on , , and , we have where and are defined in (23) and (24).
Proof. Since
By (5), (15), and (51), we have
By Taylor’s expansion, Lemma 3 and (52), we obtain
Therefore, by (54)–(56), we have
by choosing , . Theorem 7 is proved.
Finally, we study the rate of convergence of with the help of functions of Lipschitz class , where is a positive constant, . A function belongs to if
We have the following theorem.
Theorem 8. Let , , and , we have where is defined in (24).
Proof. Since and are linear positive operators, using Hölder’s inequality, we have Thus, Theorem 8 can be obtained by (24).
4. Numerical Examples
In this section, we give several numerical examples to show the convergence of and to with different values of parameters.
Example 9. Let and . The graphs of and with different values of parameters are shown in Figure 1.
Example 10. Let and . The graphs of , and for and are given in Figure 2. The error graphs of and for and are given in Figure 3. Moreover, in Table 1, there are given the maximum errors of , , and with different values of parameters, where .
5. Conclusion
In the present paper, we proposed a class of Durrmeyer type -Bernstein operators based on (, )-calculus. Due to the parameter , we have more flexibility in modeling. We studied the Korovkin type theorem, the estimated rate of convergence by using Peetres -functional, the modulus of continuity of second order and Steklov mean; we also obtained a convergence theorem for the Lipschitz continuous functions. To make things more intuitive, we also give some numerical examples.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
We declare that there is no conflict of interest.
Acknowledgments
This work is supported by the Natural Science Foundation of Fujian Province of China (Grant No. 2020J01783) and the Program for New Century Excellent Talents in Fujian Province University. We also thank Fujian Provincial Key Laboratory of Data-Intensive Computing, Fujian University Laboratory of Intelligent Computing and Information Processing, and Fujian Provincial Big Data Research Institute of Intelligent Manufacturing of China.