On the Approximation Properties of Modified Bernstein–Kantorovich Operators and Their Subfamilies

Abstract

Motivated by the construction of Bernstein–Kantorovich operators preserving affine functions, we introduce a Bernstein–Kantorovich type operators ${𝒯}_{n,m}$ together with its subfamilies ${{𝒯}^{*}}_{n,m}$, ${{𝒯}^{*}}_{n,2m-1}$ and ${{𝒯}^{*}}_{n,2m}$. We investigate and compare their approximation properties with the other Bernstein–Kantorovich operators. Our analysis demonstrates that, proposed operators, especially ${{𝒯}^{*}}_{n,2m-1}$, have significant approximation properties and preserve affine functions. Furthermore, it is shown that, for our Bernstein Kantorovich type operators ${{𝒯}^{*}}_{n,m}$, ${{𝒯}^{*}}_{n,m}(t^i,x)\to B_n(t^i,x)$, as $m\to \infty$ for $i=1,2,...$. In addition, some Voronoskaja-type theorems are established for the proposed operators. We also observe that, the operators ${{𝒯}^{*}}_{n,m}$ possesses shape-preserving properties. In the final section, the obtained results are illustrated and supported by graphical representations.