Approximation Properties of Generalized -Bernstein–Stancu-Type Operators
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.
The Bernstein operators, which are positive linear operators, are of great importance for the theory of approximation. In , 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 , 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  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 , 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 , 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 , 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 , and they considered the approximation properties and asymptotic-type results.
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 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 Pee