A Novel Class of the Modified Bernstein Operators for Functions Involving Two Variables

ABSTRACT In this paper, we introduce a new and modified class of Bernstein‐type operators defined for functions of two variables. These operators are constructed by incorporating squared central terms into the probabilistic structure of the classical Bernstein framework, enabling more refined approximation properties. We rigorously investigate the fundamental characteristics of these modified bivariate operators, including their moments, central moments, and their behavior on test functions. Our primary contributions include the establishment of various approximation theorems such as the Korovkin‐type theorem, the Voronovskaya‐type asymptotic formula, and a Grüss‐Voronovskaya‐type inequality. Furthermore, we derive quantitative estimates for the rate of convergence using tools such as the modulus of continuity, Lipschitz‐type conditions, and Peetre's ‐functional. The theoretical results are complemented by graphical and numerical examples that illustrate the approximation behavior of the proposed operators under different configurations of the parameters. This novel operator class not only generalizes classical bivariate Bernstein operators but also provides improved convergence properties, particularly for smooth bivariate functions. The results presented herein contribute to the deeper understanding of multivariate approximation theory and pave the way for potential applications in numerical analysis, computer‐aided geometric design, and surface fitting.