Abstract
In this study, we introduce a Durrmeyer type of Bleimann, Butzer, and Hahn operators (BBH) on -integers. We derive the some approximation properties for these operators. We also give some graphs and numerical examples to illustrate the convergence properties of these operators to certain functions.
1. Introduction
In the literature, especially -calculus in approximation theory is an important area and there are some applications of -calculus in this theory (see [1–4]). Therefore, the researchers in the area of approximation theory put a particular importance to -calculus. After that, the results in approximation theory related to -calculus have been extended to the results in -calculus. The researchers have presented the -extensions of linear positive operators (see [5, 6]) and they have also begun to use these operators for applying in many different areas of science (see [7–9]).
We first give some notations related to -calculus as in [10–13]. For and each natural number , the -integer of denoted by -factorial denoted by was defined byandrespectively. Also, the binomial representation was given bywhere and In [1], -beta function was defined bywhereWe have for the special case of and . Through formula (6), in [14], the authors define the -beta function which is a generalization of as follows:Now we note that the definitions of -integral are as follows: for an arbitrary function in [15]. The BBH operators have been studied in [16–24] in the approximation theory. For this reason, we define a new -Durrmeyer generalization of the BBH operators and obtain the approximation theorems involving these operators.
2. Durrmeyer Type of -BBH Operators
There are many papers mentioned about Durrmeyer type positive linear operators, such as [25–28] and so on. The Durrmeyer type of -BBH operators can be introduced via equations (1), (2), (3), (4), (5), and (8) as follows:whereandLet us give the following lemma to evaluate the values of the test functions.
Lemma 1. If , , and for integer with , then, we get the following result:
Proof. Firstly, let us give the proof of Writing instead of , we have Substitute , and use and in the following integral; we get the result And then by (8) and the equation respectively, we get the proof of Secondly, let us give the proof of Writing instead of , we have Now using we get And then by (8) and the equation respectively, we get the proof of : Lastly, let us give the proof of Writing instead of , we have Now using in the following integral, we get the result And then by (8) and the equation respectively, we get the proof of :
Lemma 2. The following results for the Durrmeyer type of -generalization of the BBH operators are verified:(i),(ii),(iii)
Proof. (i) Firstly, let us give the proof of (i). Using (10), (3), and Lemma 1, we get the following result: (ii) Secondly, let us prove (ii). From (10) and Lemma 1, we haveThus from (3), we have the desired relation in (ii).
(iii) Lastly, let us prove (iii). We use (10) and Lemma 1 to get Using the relation in the above equation, we reachUpon necessary arrangments that have been done, we haveFinally, using (3), we derive the desired relation:which completes the proof.
Remark 3. The following result is valid when , , , , and as :
3. Genuine Type of -Durrmeyer BBH Operators
Now, we define the genuine type of Durrmeyer BBH operators denoted by for as follows:wherefor
We first give the following lemma for the test functions.
Lemma 4. For , , and , where , the operators satisfy the following properties: (i),(ii),(iii)
Lemma 5. The operators satisfy the following inequality for :
4. Approximation Properties of the Operators
In this section, we will consider the space of all bounded and continuous functions on the interval denoted by and use the supremum norm for Let be the space of all-real valued functions defined on satisfying the following condition:where any and is the modulus of continuity in [22] satisfying the following conditions:(i),(ii),(iii),(iv)
Readers might see [29] for details. Now, we remember the Korovkin theorem for the linear positive operators acting from into in [22].
Theorem 6 (see [22]). If the linear positive operators acting from into satisfy the conditions, for , then we get for any function
Theorem 7. Assume that and are two sequences with , , , and as Then the assertion holds for any function .
Proof. Considering the operators, we haveIt is clear that and Using Lemma 5, we have the following inequality: Thus by (47), we get Applying Theorem 6 for the operators , we have for any function Hence using (47), we obtain Thus, we finish the proof of Theorem 7.
Theorem 8. Assume that and are two sequences with , , , , and as For the operators , we get the following estimation property: where for each and
Proof. Using , we can writeFrom the properties of the modulus of continuity, we haveFrom (54) and (55), we get Applying Cauchy-Schwarz inequality, we have Choosing as in (53) we obtain the result of Theorem 8.
From [30] we give asthe definition of general Lipschitz-type maximal functions on , where is a bounded and continuous function on , is a positive constant, , The distance of and is defined by
Theorem 9. Assume that and are two sequences satisfying the conditions , , , , and as Then for each we get the following inequality: for each and
Proof. The closure of the set is denoted by Then we have such that for Thus we have Using and linear and positive operator , we getUsing the well-known inequality for , for and , we have Thus we can write Because using Hölder inequality, we get Using (63), the proof of Theorem 9 is completed.
5. Numerical Results
In this section, we will analyze the theoretical results presented in the previous sections by numerical examples.
Example 1. Let , the graphs of with , , and are shown in Figure 1. In Figure 2, fix and let , ; the curves of (the black curve, denoted by in Figure 2) and (the red curve) are shown; let , ; the curves of (the purple curve, denoted by in Figure 2) and (the brown curve) are shown; let , ; the curves of (the green curve, denoted by in Figure 2) and (the brown curve) are shown. In Figure 3, the graphs of and with , , and are shown. Tables 1 and 2 are the absolute error bound of to and to with different values of and . One can see from Tables 1 and 2, the latter operators are better than the former ones .
Data Availability
The data used to support the findings of this study are included within the article.
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 no. 11601266), the Natural Science Foundation of Fujian Province of China (Grant no. 2016J05017), the Project for High-level Talent Innovation and Entrepreneurship of Quanzhou (Grant no. 2018C087R), 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.