In the present paper, Durrmeyer type -Bernstein operators via (, )-calculus are constructed, the first and second moments and central moments of these operators are estimated, a Korovkin type approximation theorem is established, and the estimates on the rate of convergence by using the modulus of continuity of second order and Steklov mean are studied, a convergence theorem for the Lipschitz continuous functions is also obtained. Finally, some numerical examples are given to show that these operators we defined converge faster in some cases than Durrmeyer type (, )-Bernstein operators.

中文翻译：

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关于通过（，）-微积分的Durrmeyer型-Bernstein算子

在本文件中，Durrmeyer型-经由Bernstein算（， ） -演算构造中，第一和第二时刻和这些运算符的中心矩估计，一个Korovkin型逼近定理成立，并且在收敛的由速率估计利用二阶连续模和Steklov均值，研究了Lipschitz连续函数的收敛定理。最后，通过一些数值例子表明，在某些情况下，我们定义的这些算子收敛速度比Durrmeyer型（， ）- Bernstein算子快。