Abstract
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.
Highlights
A first fundamental result in approximation theory was Weierstrass approximation theorem [1]which forms the solid foundation of Approximation Theory
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: (θ )
We have introduced generalized Bernstein–Durrmeyer type operators depending on non-negative integers
Summary
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. There were several proofs given by different famous mathematicians. One of them was given by Bernstein [2] which was easy and elegant, which motivated the researchers to construct operators to deal with the approximation problems in different settings. 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:
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