Higher order \(\alpha\)-Bernstein-Kantorovich operators (Q6584780)

Bernstein polynomials \(B_n(\xi,\kappa)=\sum_{i=0}^n\xi(\frac{i}{n})P_{n,i}(\kappa),\ P_{n,i}(\kappa)=\binom{n}{i}\kappa ^i(1-\kappa)^{n-i},\ \kappa \in[0,1]\) play a significant role in approximation theory. To improve the convergence properties of these polynomials, first \(P_{n,i}\) was replaced by depending on some parameter \(\alpha\) polynomials \(P_{n,i}^\alpha\) coinciding with \(P_{n,i}\) in the case \(\alpha =1\). The resulting polynomials were later modified by L. Kantorovich. This modification is based on that the simple value \(\xi(\frac{i}{n})\) is replased with the mean values of \(\xi\) calculated within the \([\frac{k}{n+1},\frac{k+1}{n=1}]\) intervals. Then, after performing \(l^{th}\) order integration, we obtain a new family of operators \(K_{n,\alpha}^{l,\alpha}\) called as \(\alpha\)-Bernstein-Kantorovich operators. For these operators it is analyzed various aspects, such as error estimations and Voronovskaja-type asymptotic formula. Finally, the convergence of the operator \(K_{n,\alpha}^{l,\alpha}\) is examined applied to a certain functions. A comparative study by visualizing the results graphically is provided.