In this work, we extend the works of F. Usta and construct new modified -Bernstein operators using the second central moment of the -Bernstein operators defined by G. M. Phillips. The moments and central moment computation formulas and their quantitative properties are discussed. Also, the Korovkin-type approximation theorem of these operators and the Voronovskaja-type asymptotic formula are investigated. Then, two local approximation theorems using Peetre’s -functional and Steklov mean and in terms of modulus of smoothness are obtained. Finally, the rate of convergence by means of modulus of continuity and three different Lipschitz classes for these operators are studied, and some graphs and numerical examples are shown by using Matlab algorithms.

中文翻译：

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-Bernstein 算子在 (0,1) 上的新修改的近似定理

在这项工作中，我们扩展了 F. Usta 的工作，并使用GM Phillips 定义的- Bernstein 算子的第二个中心矩构造了新的修正- Bernstein 算子。讨论了矩和中心矩计算公式及其定量性质。此外，还研究了这些算子的 Korovkin 型近似定理和 Voronovskaja 型渐近公式。然后，获得了使用 Peetre's -泛函和 Steklov 均值以及平滑模量的两个局部逼近定理。最后，研究了这些算子通过连续性模数和三个不同的Lipschitz类的收敛速度，并使用Matlab算法给出了一些图形和数值例子。