Novel Bernstein-Kantorovich Type Operators

Abstract

In this paper, we introduce a novel class of Bernstein-Kantorovich operators, denoted as $K_{\beta }^{n,\theta }$, parameterized by and $\theta >0$. We first derive a recurrence formula that facilitates the computation of moments and central moments and provide explicit expressions for $K_{\beta }^{n,\theta }(t^m;x)$ and $K_{\beta }^{n,\theta }({(t-x)}^m;x)$ for $m=0,1,2,3,4$. A comparative analysis reveals that, under specific conditions on $\beta$ and $\theta$, the second-order central moments of our new operators are smaller than those of the classical Kantorovich operators, indicating superior convergence. We further establish that these operators preserve fundamental shape properties such as monotonicity and convexity. A Korovkin-type theorem is proved, ensuring uniform convergence for continuous functions. We also present local approximation theorems using the first- and second-order modulus of continuity, including an estimate via a Lipschitz-type maximal function, and a global direct approximation result in terms of the Ditzian-Totik modulus of the second order. Furthermore, we prove both qualitative and quantitative Voronovskaja-type theorems, which describe the asymptotic behavior of the approximation error. Finally, a numerical convergence analysis is conducted to validate the theoretical results and identify optimal parameter choices.