In this paper, we construct a Durrmeyer variant of the modified \(\alpha \)-Bernstein-type operators introduced by Kajla and Acar (Ann Funct Anal 10(4):570–582, 2019), for \(\alpha \in [0,1]\). We investigate the degree of approximation via the approach of Peetre’s K-functional and the Lipschitz-type maximal function. The quantitative Voronovskaja- and Grüss Voronovskaja-type theorems are discussed. Further, we determine the rate of convergence by the above operators for the functions with derivatives of bounded variation.

中文翻译：

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修改后的 $$\alpha $$ α -Bernstein–Durrmeyer 类型运算符

在本文中，我们构建了由 Kajla 和 Acar（Ann Funct Anal 10(4):570–582, 2019）引入的修正\(\alpha \) -Bernstein 型算子的 Durrmeyer 变体，对于\(\alpha \在 [0,1]\) 中。我们通过 Peetre 的K泛函和 Lipschitz 型极大函数的方法来研究逼近程度。讨论了定量 Voronovskaja 和 Grüss Voronovskaja 型定理。此外，我们通过上述算子确定具有有界变化导数的函数的收敛速度。