Abstract

In this manuscript, we consider a bivariate extension of modified ρ-Bernstein operators and obtain Voronovskaya type and Grüss Voronovskaya type theorems for these operators. Further, we determine the rate of convergence of these operators in terms of the complete and partial moduli of continuity and compute an estimate of the error in terms of the Peetre’s K-functional. Also, we define the associated Generalized Boolean Sum (GBS) operators and study the rate of convergence of these operators with the aid of the mixed modulus of smoothness for the Bögel continuous and Bögel differentiable functions and the degree of approximation for the Lipschitz class of Bögel continuous functions.

摘要

在这份手稿中，我们考虑了修正的 ρ-Bernstein 算子的二元扩展，并获得了这些算子的 Voronovskaya 型和 Grüss Voronovskaya 型定理。此外，我们根据连续性的完全模和部分模确定这些算子的收敛速度，并根据 Peetre 的 K 泛函计算误差估计。此外，我们定义了相关的广义布尔和 (GBS) 算子，并借助 Bögel 连续和 Bög​​el 可微函数的混合平滑模数和 Bög​​el 的 Lipschitz 类的近似度研究这些算子的收敛速度连续函数。