Quantitative Voronovskaya type theorems and GBS operators of Kantorovich variant of Lupaş-Stancu operators based on Pólya distribution

The motivation behind the current paper is to elucidate the approximation properties of a Kantorovich variant of Lupaş-Stancu operators based on Pólya distribution. We construct quantitative-Voronovskaya and Grüss-Voronovskaya type theorems and determine the convergence estimates of the above operators. We also contrive the statistical convergence and talk about the approximation degree of a bivariate extension of these operators by exhibiting the convergence rate in terms of the complete and partial moduli of continuity. We build GBS (Generalized Boolean Sum) operators allied with the bivariate operators and estimate their convergence rate using mixed modulus of smoothness and Lipschitz class of B\begin{document}$ \ddot{o} $\end{document}gel continuous functions. We also evaluate the order of approximation of the GBS operators in the spaces of B-continuous (Bögel continuous) and B-differentiable (Bögel differentiable) functions. In addition, we depict the comparison between the rate of convergence of the proposed bivariate operators and the corresponding GBS operators for some functions by graphical illustrations using MATLAB software.