In this article, we purpose to study some approximation properties of the one and two variables of the Bernstein-Schurer-type operators and associated GBS (Generalized Boolean Sum) operators on a symmetrical mobile interval. Firstly, we define the univariate Bernstein-Schurer-type operators and obtain some preliminary results such as moments, central moments, in connection with a modulus of continuity, the degree of convergence, and Korovkin-type approximation theorem. Also, we derive the Voronovskaya-type asymptotic theorem. Further, we construct the bivariate of this newly defined operator, discuss the order of convergence with regard to Peetre’s -functional, and obtain the Voronovskaya-type asymptotic theorem. In addition, we consider the associated GBS-type operators and estimate the order of approximation with the aid of mixed modulus of smoothness. Finally, with the help of the Maple software, we present the comparisons of the convergence of the bivariate Bernstein-Schurer-type and associated GBS operators to certain functions with some graphical illustrations and error estimation tables.

中文翻译：

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对称移动区间上Bernstein-Schurer型算子和相关GBS算子的一个和两个变量的逼近

在本文中，我们旨在研究对称移动区间上Bernstein-Schurer型算子和关联的GBS（广义布尔和）算子的一个和两个变量的一些近似性质。首先，我们定义单变量Bernstein-Schurer型算子，并获得一些初步结果，例如矩，中心矩，以及连续模量，收敛度和Korovkin型逼近定理。此外，我们推导了Voronovskaya型渐近定理。此外，我们建立这个新定义的运算符的二元，对于Peetre的讨论收敛的顺序-函数，并获得Voronovskaya型渐近定理。此外，我们考虑了相关的GBS类型算子，并借助混合平滑系数来估计近似阶数。最后，借助Maple软件，我们通过一些图形化图示和错误估计表，比较了双变量Bernstein-Schurer型和关联的GBS算符对某些函数的收敛性。