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 V^*n,m, along with the associated subfamilies V^*n,m, V^*n,2m-1, and V^*n,2m. The approximation behavior of these operators is thoroughly investigated and compared with that of classical Baskakov--Kantorovich operators Vₙ. Our results demonstrate that the proposed operators, Particularly V^*n,2m-1 exhibit superior approximation properties while preserving affine functions. Moreover, we establish that for the operators V^*n,m, the moments satisfy\[V^*n,m(t^i,x) → V_n(t^i,x) as m → ∞, i=1,2,…\]Several Voronovskaja-type theorems are also derived for the newly introduced operators. In addition, we show that the operators V^*n,m possess important shape-preserving properties. Finally, numerical examples and graphical illustrations are presented to validate and visualize the theoretical findings.