In this work, we construct a new kind of Bernstein–Schurer operators which includes non-negative real parameter $$\alpha $$ . We study some shape preserving properties, namely, monotonicity and convexity of the new operators. We obtain global approximation formula in terms of Ditzian–Totik uniform modulus of smoothness of first and second order and calculate the local direct estimate of the rate of convergence with the help of Lipschitz-type function for our operators. The Voronovskaja-type approximation theorems of the new operators are presented. Finally, in the last section, we provide some graphs, MATLAB code and numerical examples in order to illustrate the significance of our newly constructed operators.

中文翻译：

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通过一类新的广义 Bernstein-Schurer 算子逼近函数

在这项工作中，我们构造了一种新的 Bernstein-Schurer 算子，其中包括非负实参数 $$\alpha $$ 。我们研究了一些形状保持特性，即新算子的单调性和凸性。我们根据一阶和二阶平滑度的 Ditzian-Totik 均匀模数获得全局近似公式，并在我们的算子的 Lipschitz 型函数的帮助下计算收敛率的局部直接估计。介绍了新算子的 Voronovskaja 型近似定理。最后，在最后一节中，我们提供了一些图表、MATLAB 代码和数值示例，以说明我们新构造的运算符的重要性。