The prime objective of this paper is to construct the Kantorovich variant of Ismail–May operators depending on a non-negative parameter \(\lambda \). We estimate the rate of convergence of the proposed operators for functions in Lipschitz-type space. Further, an improved quantitative Voronovskaya-type estimate in terms of the second-order modulus of continuity and a direct approximation theorem using Ditzian–Totik modulus of smoothness is also given. The last section is dedicated to the bivariate generalisation of the proposed operators and estimation of their rate of convergence in terms of partial and total modulus of continuity and Peetre’s K-functional. A Voronovskaya-type result is also obtained. Finally, some graphs and error estimation table to illustrate the convergence of the proposed operators are presented using Mathematica software.

中文翻译：

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Ismail–May操作员的Kantorovich变体

本文的主要目的是根据非负参数\（\ lambda \）构造Ismail–May运算符的Kantorovich变体。我们估计Lipschitz型空间中拟议的算子对函数的收敛速度。此外，还给出了基于二阶连续模量的改进的定量Voronovskaya型估计和使用Ditzian-Totik平滑模量的直接近似定理。最后一部分专门讨论拟议算子的二元概括，并根据部分和总连续模数以及Peetre的K估计其收敛速度功能。还获得了Voronovskaya型的结果。最后，使用Mathematica软件提供了一些图表和误差估计表，以说明所提议算子的收敛性。