Voronovskaja Type Theorems for King Type Operators

Abstract

We introduce and study the King type operators associated to a couple $\left( \mathcal {A},\tau \right)$ where $\mathcal {A}=\left( A_{n}\right) _{n\in \mathbb {N}}$ is a sequence of linear positive operators from $C\left[ 0,1\right]$ into $C\left[ 0,1\right]$ and $\tau :\left[ 0,1\right] \rightarrow \left[ 0,\infty \right)$ a continuous strictly increasing function. Given a sequence $\Lambda =\left( \lambda _{n}\right) _{n\in \mathbb {N}}$ with $\lim \nolimits _{n\rightarrow \infty }\lambda _{n}=\infty$ we introduce the concept of the $\Lambda$-Voronovskaja property of a function $f\in C\left[ 0,1\right]$ with respect to the sequence $\mathcal {A}$. We show that there is a natural connection between the $\Lambda$-Voronovskaja property with respect to the sequence $\mathcal {A}$ and the $\Lambda$-Voronovskaja property with respect to the sequence of King type operators. We apply these general results to the case of Bernstein, Kantorovich type operators and thus obtain entirely new Voronovskaja type theorems for such a kind of positive linear operators.