Approximation by <i>α</i> -Bernstein-Kantorovich Type Operators for Functions of One and Two Variables (Q6657540)

A sequence of summation-integral type operators was introduced by \textit{N. K. Govil} et al. [Appl. Math. Comput. 225, 195--203 (2013; Zbl 1334.41029)], who also examined their approximation results. \textit{S. A. Mohiuddine} et al. [Math. Methods Appl. Sci. 40, No. 18, 7749--7759 (2017; Zbl 1387.41008)] introduced the modified Bernstein-Kantorovich operators depend on \(\alpha\) and established the uniform convergence as well as the rate of convergence of the operators. \textit{A. M. Acu} et al. [Carpathian J. Math. 38, No. 1, 1--12 (2022; Zbl 1538.41039)] generalized the exponential Bernstein Kantorovich type operators and estimated approximation properties. \textit{D. Costarelli} et al. [Math. Nachr. 296, No. 2, 588--609 (2023; Zbl 1540.41032)] considered Durrmeyer type modification of sampling series in Orlicz space. \textit{A. Senapati} et al. [Rend. Circ. Mat. Palermo (2) 72, No. 7, 3749--3764 (2023; Zbl 1530.41016)] proposed modified Bernstein Kantorovich type operators and evaluated approximation behavior. \textit{A. Kajla} and \textit{M. Goyal} [Rend. Circ. Mat. Palermo (2) 67, No. 2, 379--395 (2018; Zbl 1396.26004)] established approximation results for single and bivariate modified Bernstein-Kantorovich type operators. \textit{S. Begen} and \textit{H. G. İlarslan} [Honam Math. J. 42, No. 2, 251--268 (2020; Zbl 1476.41003)] constructed bivariate Szász-Kantorovich type operators using Brenke-type polynomials. \textit{A. M. Acu} and \textit{H. Gonska} [Ukr. Math. J. 71, No. 6, 843--852 (2019; Zbl 1435.41021)] derived a quantitative Voronovskaja type theorem in terms of second modulus of smoothness that enhances certain known estimates for classical Bernstein-Kantorovich operators. \textit{X. Chen} et al. [J. Math. Anal. Appl. 450, No. 1, 244--261 (2017; Zbl 1357.41015)] proposed a new class of the Bernstein operators known as \(\alpha\)-Bernstein operators. For \(\alpha \in [0, 1]\), \textit{M. Sofyalıoglu} et al. [Filomat 36, No. 5, 1699--1709 (2022; \url{doi:10.2298/FIL2205699S})] presented modified Bernstein operators involving \(\alpha\).\N\N\NIn this paper under review, the authors introduce a new sequence of \(\alpha\)-Bernstein-Kantorovich type operators based on a parameter \( \zeta\) to study approximation behavior of the operators, They establish local and global approximation theorems. They also introduce bivariate \(\alpha\)-Bernstein-Kantorovich operators and estimate the Voronovskaya type asymptotic theorem and the order of approximation using Peetre's \(K\)-functional. Using Maple software, they show the convergence of the newly defined operators to a certain function by graph.