Durrmeyer-Type Generalization of Parametric Bernstein Operators

: In this paper, we present a Durrmeyer type generalization of parametric Bernstein operators. Firstly, we study the approximation behaviour of these operators including a local and global approximation results and the rate of approximation for the Lipschitz type space. The Voronovskaja type asymptotic formula and the rate of convergence of functions with derivatives of bounded variation are established. Finally, the theoretical results are demonstrated by using MAPLE software.


Introduction
A first fundamental result in approximation theory was Weierstrass approximation theorem [1] which forms the solid foundation of Approximation Theory. The proof of the theorem was quite long and difficult. So there were several proofs given by different famous mathematicians. One of them was given by Bernstein [2] which was easy and elegant, which also motivated the researchers to construct operators to deal with the approximation problems in different settings. Here, we give a Durrmeyer type generalization of parametric Bernstein operators.Let C() be the space of all real valued continuous functions S on the interval  = [0, 1]. For S ∈ C(), Chen et al. [3] introduced a new family of generalized Bernstein operators depending upon a non-negative real parameter 0 ≤ θ ≤ 1, which is given as follows: where If ρ 1 = ρ 2 = 0, these operators reduces to the operators T (θ) m . For S ∈ C(), we introduce a Durrmeyer type modification of the operators (2) as follows: The aim of this paper is to derive approximation properties for the operators (3) by working on Korovkin's results [14]. We also compute the rate of convergence involving modulus of smoothness and Lipschitz type function.

Auxiliary Results
In this section, we derive some auxiliary results which will be used in proving our main results of subsequent sections. First, we determine moments and central moments for the operators (3). Lemma 1. Let e i (t) = t i , i = 0, 1, 2 · · · . For the operators U (θ) m,ρ 1 ,ρ 2 , we have
Proof. This lemma is established by direct computation and the details are missing.

Remark 1. For the operators U
Proof. From Lemma 1 and Equation (3), we obtain

Voronovskaja Type Theorems
Here, we establish the Voronovskaja, Grüss-Voronovskaja type theorems and related results.
Proof. Applying the application of Taylor's theorem, we have which implies that lim This complete the first half of the theorem.
To show the uniformity postulation, by the definition of uniformly continuity of S in , the δ must be independent of x and all the other estimates hold uniformly in x ∈ .
In [15], Acar et al. obtained a Grüss type approximation result and a Grüss-Voronovskaja-type result for linear and positive operators. Many authors have established in this direction so that we refer the authors to [16][17][18] and references therein.

Local Approximation
In this section, we study the local approximation property for our operators with the help of K-functional.
The K-functional is given by : where W 2 = {h :h ∈ C()} and uniform norm on C() is denoted by ||.||. By [20] there will be a positive constant M > 0 such that where the second order modulus of continuity for S ∈ C() is defined as We define the usual modulus of continuity for S ∈ C() as Proof. We define the auxiliary operators as follows: Then, we can easily check that By the application of Taylor's theorem and taking t ∈  andh ∈ W 2 , we get The operator U (θ) m,ρ 1 ,ρ 2 is applied in the above equation on both sides, we obtain From Lemma 3, we have Thus, by (8) we have where x ∈ . Furthermore, by Lemma 4, we have for all S ∈ C() and x ∈ .
Now, for S ∈ C() andh ∈ W 2 , using (10) and (11) we obtain that Using the property of infimum on the right hand side over allh ∈ W 2 , we have Now by examining the relation (7), we get

Global Approximation
The following result provides the global approximation using the modulus of continuity of Ditzian-Totik and the related K-functional.
Suppose that S ∈ C() and ϕ(x) is defined as x(1 − x), x ∈ . The second order modulus of continuity which is given by Ditzian-Totik and related K-functional is defined as, where W 2 (ϕ) = {h ∈ C() :h ∈ AC loc , ϕ 2h ∈ C()} andh ∈ AC loc  means thath is derivable and h is absolutely continuous on every closed interval [a, b] ⊂ (0, 1). By ( [21],Theorem 1.3.1) we can say that ∃ M > 0, such thatK The first order Ditzian-Totik modulus is defined as where ψ :  → R is an admissible step-weight function.
Now we state our next main theorem.

Rate of Approximation
In this section, we study the rate of convergence of functions with derivatives of bounded variation.
The class of all absolutely continuous functions S is denoted by DBV () , defined and having a derivative S on , analogous to a bounded variation function on .
The representation of functions S ∈ DBV () is whereh is a bounded variation function on .
The operators U (θ) m,ρ 1 ,ρ 2 (S; x) also admit the integral representation where the kernel N (θ) m,ρ 1 ,ρ 2 (x, t) is given by Lemma 5. For a fixed x ∈ (0, 1) and sufficiently large m, we have The (ii) can be proved in the same way hence the details are skipped.
Theorem 8. Suppose that S ∈ DBV(). Then for every x ∈ (0, 1) and sufficiently large m, we have where d c (S x ) denotes the total variation of S x on [c, d] and S x is defined by Proof. This theorem can be proved in the same way as in ([4], Theorem 7). Hence, the proof of this theorem is skipped.

Numerical Examples
In the following examples, we demonstrate the theoretical results by graphs.

Conclusions
We have introduced generalized Bernstein-Durrmeyer type operators depending on non-negative integers. We developed many approximation properties such as local and global approximation, the rate of approximation for the Lipschitz type space, Voronovskaja type asymptotic formula and the rate of convergence of functions with derivatives of bounded variation. The constructed operators have better flexibility and rate of convergence which are depending on the selection of the ρ 1 , ρ 2 and θ. Graphical representations of our operators for different selections of ρ 1 , ρ 2 and θ are also given.