Abstract
The present study introduces generalized -Bernstein–Stancu-type operators with shifted knots. A Korovkin-type approximation theorem is given, and the rate of convergence of these types of operators is obtained for Lipschitz-type functions. Then, a Voronovskaja-type theorem was given for the asymptotic behavior for these operators. Finally, numerical examples and their graphs were given to demonstrate the convergence of to with respect to values.
1. Introduction
The Bernstein operators, which are positive linear operators, are of great importance for the theory of approximation. In [1], Bernstein operators were introduced by Bernstein to prove the Weierstrass approximation theorem. For , the classical Bernstein operators are given aswhere the Bernstein basis functions are defined as
There are many generalizations of the operators. In [2], Gadjiev and Ghorbanalizadeh gave Bernstein–Stancu operators with shifted knots:where and for , and are positive real numbers, and for . In the case of for , operator (3) reduces to the classical Bernstein operator (1). In addition, the case of was handled and examined by Stancu in [3] and these operators were called the classical Bernstein–Stancu operators. A Durrmeyer variant of the Bernstein–Stancu operators in (3) was studied by Dinlemez Kantar and Ergelen in [4], and a Voronovskaja-type approximation theorem for these operators was given. Several studies were conducted on some approximation properties, and asymptotic-type results were given for these operators in [5–14].
In [5], Cai et al. introduced the Bernstein operators with shape parameter as follows:where and are Bézier basis functions with shape parameter defined by
Later in [6], Cai proposed Kantorovich-type -Bernstein operators, as well as their Bézier variant, and examined the approximation results. Kantorovich-type -Bernstein operators were also studied by Acu et al. in [7], and they considered the approximation properties and asymptotic-type results.
Recently, in [8], Srivastava et al. constructed -Bernstein–Stancu operators defined bywhere and Bézier basis functions with shape parameter are defined in (5).
In the present study, we introduce the following generalized -Bernstein–Stancu operators with shifted knots for :where and are positive real numbers satisfying for and Bézier basis functions with shape parameter are defined bywhere for , are positive real numbers, for , and . We give approximation properties and Voronovskaja-type approximation theorem for asymptotic behavior of operator (7). When for and , operator (7) reduces to the classical Bernstein operator (1). When , it reduces to the Bernstein–Stancu operator (3). When for , it reduces to the Bernstein operator with shape parameter (4). In the following section, several lemmas are given to prove the main results.
2. Some Preliminary Results
Lemma 1. For generalized -Bernstein–Stancu operators with shifted knots, we have the following equalities:
Proof. If we use Bézier basis functions (8) in -Bernstein–Stancu operators (7), we obtainwhereThanks to the linearity of Bernstein–Stancu operators (3), we obtainNow, we will compute and Using (12), , and in , we obtain the following equation:Then, we have the following equality for the third moment by using the linearity of :whereFrom the linearity of the Bernstein–Stancu operators (3)and Bernstein basis functions, we obtainNext, we compute and Combining (17), , and , we obtain the result for
Using the same technique in the above moments, we obtain and
Corollary 1. Using Lemma 1, we obtain the following inequalities of central moments for and for fixed and :Using Lemma 1 and the linearity of , we have the following Corollary 2.
Corollary 2. We obtain the following equalities:
3. Convergence Properties of
For the asymptotic behavior of operators, we give the following Korovkin-type approximation theorem.
Theorem 1. If and , then operators converge uniformly to on , where is a Banach space of all continuous functions on with norm .
Proof. Using the equalities , and of Lemma 1, we obtainTherefore, the proof is completed using Korovkin theorem.
For , the Peetre K-functional is defined aswhere . For , the second-order modulus of continuity is defined asBy [9, Theorem 2.4], there exists an absolute constant such thatAlso, the usual modulus of continuity of is defined as .
Theorem 2. We obtain the following inequality for and :where and are defined in Corollary 1.
Proof. Let us define the following operators:From the linearity of and the equalities and of Lemma 1, we obtainUsing Taylor’s expansion for , we writeApplying generalized -Bernstein–Stancu operators to both sides of (27) and using (26), we yieldAnd, we findTherefore, if we take infimum on the right side of (29), overall , we obtainUsing inequality (23), we haveThus, Theorem 2 is proved.
Remark 1. Since and for , these limits give us a rate of pointwise convergence of the operators to .
The space of the Lipschitz-type functions is defined aswhere and [10].
In the following theorem, we obtain the rate of convergence of generalized -Bernstein–Stancu operators for functions in .
Theorem 3. If , , and , then we havewhere is defined in Corollary 1.
Proof. Because and are linear positive operators, we obtain the inequality by using Hölder’s inequality:Hence, we proved Theorem 3.
Finally, we give the main result of the article in Theorem 4.
Theorem 4. If , then for every and , we obtainwhere exists.
Proof. Using Taylor’s formula, we can write the following equation for a fixed :where is Peano form of the remainder, , and . If we apply to (36), then we haveand we obtainUsing Cauchy–Schwarz inequality, we obtainBecause and are finite operators, we haveIn the end, by using the equalities and of Corollary 2 and (40) in (38), we yieldThus, we proved Theorem 4.
4. Numerical Examples
In this section, we show the theoretical results demonstrated in the previous sections by the following example.
Example 1. Let the trigonometric function for , , and . First, let us choose and .
For , the graphs of and are shown in Figure 1.
For , the graphs of and are shown in Figure 2.
Now, let us choose and . The graph of and the graph of with and are shown in Figures 3 and 4, respectively.
As a result, Figure 5 reveals that the curve of with and studied in the article approaches the curve of the function much better than the curves of the operators and defined in [5, 8], respectively.
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
The authors would like to thank the 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. This work was supported by the Natural Science Foundation of Fujian Province of China (Grant no. 2020J01783), the Project for High-Level Talent Innovation and Entrepreneurship of Quanzhou (Grant no. 2018C087 R), and the Program for New Century Excellent Talents in Fujian Province University.