Chlodowsky-Type Variants of Agratini Operators: Approximation Properties and Bivariate Extension

In this study, we introduce a Chlodowsky-type generalization of the linear positive Agratini operators and investigate their fundamental approximation properties. We derive exact algebraic expressions for the moments and central moments of the proposed operators. Furthermore, we establish the rate of convergence and obtain quantitative estimates of the degree of approximation using the classical modulus of continuity on expanding intervals. We also investigate the local approximation properties for Lipschitz classes and establish a standard Voronovskaya-type asymptotic theorem. Additionally, we construct a bivariate extension of the newly introduced operators and investigate its weighted convergence properties and rates of convergence. Finally, we present numerical evaluations and graphical illustrations to support the theoretical convergence results.