Generalized Baskakov-Kantorovich Operators with Better Approximation

Motivated by the development of Bernstein--Kantorovich operators that preserve affine functions, we propose a new class of Baskakov--Kantorovich type operators $\mathcal{V^*}_{n,m}$, along with the associated subfamilies $\mathcal{V}^*_{n,m}$, $\mathcal{V}^*_{n,2m-1}$, and $\mathcal{V}^*_{n,2m}$. The approximation behavior of these operators is thoroughly investigated and compared with that of classical Baskakov--Kantorovich operators $V_{n}$. Our results demonstrate that the proposed operators, Particularly $\mathcal{V}^*_{n,2m-1}$ exhibit superior approximation properties while preserving affine functions. Moreover, we establish that for the operators $\mathcal{V}^*_{n,m}$, the moments satisfy\[\mathcal{V}^*_{n,m}(t^i,x) \to V_n(t^i,x) \quad \text{as } m \to \infty, \quad i=1,2,\ldots\]Several Voronovskaja-type theorems are also derived for the newly introduced operators. In addition, we show that the operators $\mathcal{V}^*_{n,m}$ possess important shape-preserving properties. Finally, numerical examples and graphical illustrations are presented to validate and visualize the theoretical findings.